Skip to main content

Full text of "Elementary Nuclear Theory: Part 2"

See other formats


XIV. SATURATION OF NUCLEAR FORCES 

The binding energy and volume of nuclei are proportional to A. 

the mass number. This is not in accord a villi a, law of force which 
gives equal interactions between all pairs of particles in the nucleus 
for there are then A. [A — l)/2 distinct interacting pairs and a 
binding energy at least proportional to A (A — l)/2 might be 
expected, if not to a higher power of A. due to increased packing 
with more interaction. Instead, the nuclear binding energies 
seem similar to the internal energies of bulk matter, in which 2 
pounds has twice as much energy and volume as 1 pound. 

To account for this phenomenon of "saturation of nuclear 
forces," in which one particle apparently interacts with only a 
limited number of others, various hypotheses have been made, 
and various other assumptions about the nature of the forces can 
be shown to bo impossible. 

Among the impossible assumptions is that which has been used 
in this book so far, namely, an ordinary potential independent of 
the angular momentum, because it is easily shown that such a 
potential does not give saturation. This is so even if the Coulomb 
repulsion of the protons is taken into account. The proof can be 
carried out with various degrees of exactness, using the variational 
method. This method is based ou the Schrodinger variational 
theorem which states that the quantity 

9. = $f1B$ dr/jy dr (1-14) 

is a minimum when ^ is the correct eigenf unction of the lowest 
eigenvalue E (i of //, and the minimum value of Q is E$. Thus, it 
the assumed Hamil Ionian operator representing the interaction 
of the particles in a given nucleus is sandwiched between any 
arbitrary ip in the expression for 11, the value of must be greater 
(i.e., less negative) than the correct energy of that nucleus. The 
simplest i^'s are plane waves inside a box representing the nucleus. 
If the size of the box is adjusted to give as low an as possible, 
this size comes out about equal to the range of nuclear forces, 
which is clearly much too small. Further, it gives a potcntia 

80 



SATURATION OF NUCLEAR FORCES 



SI 



nergy proportional to A 2 , and a kinetic energy proportional to 
a# The size of the coefficients of these powers is such that the 
J otential energy dominates for A > 50 ; for .4 = 238 the binding 
energy is greater than 238 mass units. This is convincing evidence; 
that the ordinary potential will not work, and this is true inde- 
pendently of the shape of the potential (square well, exponential, 
Gaussian, etc.). 

What is needed is a potential which prevents the particles from 
getting too close together. A. poten fial repulsive at short distances, 
originally used by Morse for molecules, has been explored by 
Schiff and Fisk; the only objection is that the high repulsive 
potential may give relativistic difficulties if it gets above 
2Mc 2 ~ 1800 Mev, for a proton in such a state would have nega- 
tive kinetic energy. However, the idea of a repulsive potential 
has not been followed up sufficiently. 



EXCHANGE FORCES 

In the first paper on nuclear forces, TTeisenberg proposed, in 
order to explain the saturation of nuclear forces, that these forces 
are "exchanger" forces, similar to the force that binds ordinary 
chemical molecules. Without inquiring into the origin of these 
exchange forces, let us write down the various types of exchange 
forces that can exist between two particles, and then examine the 
effects of these forces on the properties of the deuteron, and on the 
saturation of the binding energy. 

For an ordinary (non-exchange) central force the Schrodinger 
equation for two particles is (in the center-of-mass system) : 



[(n 2 /M)V 3 + Etyixu % frj, ffjO 



7(^(^*3* <rt„vz} (W5) 
Wigner 



In nuclear physics, such forces are called Wigner forces. The 
interaction does not cause any exchange between, coordinates of 
the two particles. Another type of interaction is bile that inter- 
changes the space coordinates of the two particles in addition to 
multiplication of $ by some V(r); for such an interaction, the 
Schrodinger equation is : 

[{ii 2 /M)V 2 + Emt h t,, ffu ad = FWtffct, r,, <r u *a) (146) 

Major ana 



XIV. SATURATION OF NUCLEAR FORCES 

The binding energy and volume of nuclei are proportional to A, 
the mass number. This is not in accord with a law of force which 
gives equal interactions between all pairs of particles in the nucleus, 
for there are then A (A — J.)/2 distinct interacting pairs and a 
binding energy at least proportional to A(A — L)/2 might be 
expected, if not to a higher power of .4 clue to increased packing 
with more interaction. Instead, the nuclear binding energies 
seem similar to the internal energies of bulk matter, in which 2 
pounds has twice as much energy and volume as 1 pound. 

To account for this phenomenon of ''saturation of nuclear 
forces," in which one particle apparently interacts wdth only a 
limited number of others, various hypotheses have been made, 
and various other assumptions about the nature of the forces can 
be shown to be impossible. 

Among the impossible assumptions is that which has been used 
in this book so far, namely, an ordinary potential independent of 
the angular momentum, because ft is easily shown that such a 
potential does not give saturation. This is so even if the Coulomb 
repulsion of the protons is taken into account. The proof can be 
carried out "with various degrees of exactness, using the variational 
method. This method is based on the Schro dinger variational 
theorem which states that the quantity 



9. = f$£$ dr/jf 2 dr 



(1443 



is a minimum when ^ is the correct eigenfunetion of the lowest 
eigenvalue Eq of H, and the minimum value of V. is E$, Thus, if 
{hg assumed Ilamiltonian operator representing the interaction 
of the particles in a given nucleus is sandwiched between any 
arbitrary $ in the expression for fi, the value of must be greater 
(i.e., less negative) than the correct energy of that nucleus. The 
simplest ^'s are plane waves inside a box representing the nucleus, 
ff the size of the box is adjusted to give as low an 9. as possible, 
this size comes out about equal to the range; of nuclear forces, 
which is clearly much too small. Further, it gives a potential 

80 



SATURATION OF NUCLEAR FORCES 



SI 



energy proportional to A 2 , and a kinetic energy proportional to 
A H , The size of the coefficients of these powers is such that the 
potential energy dominates for A > 50; for A = 238 the binding 
energy is greater than 23S mass mats, This is convincing evidence 
that the ordinary potential will not work, and this is true inde- 
pendently of the shape of the potential (square well, exponential, 
Gaussian, etc.). 

What is needed is a potential which prevents the particles from 
getting too close together. A potential repulsive at short distances, 
originally used by Morse for molecules, has bees explored by 
Schiff and Fisk; the only objection is that the high repulsive 
potential may give relativlstic difficulties if it gets above 
2il/c 2 -^ 1800 Mev, for a proton in such a state would have nega- 
tive kinetic energy. However, the idea of a repulsive potential 
has not been followed up sufficiently. 



EXCHANGE FORCES 

In the first paper on nuclear forces, ITeisenberg proposed, in 
order to explain the saturation of nuclear forces, that these forces 
are ''exchange" forces, similar to the force that binds ordinary 
chemical molecules. Without inquiring into the origin of these 
exchange forces, let us write down the various types of exchange 
forces that can exist between two particles, and then examine the 
effects of these forces on the properties of the deuteron, and on the 
saturation of the binding energy, 

For an ordinary (non-exchange) central force the Schrodinger 
equation for two particles is (in the center-of-mass system) : 



[(h 2 /M)V 2 + Ety(r u r 2 , a u <r. 2 ) 



Wigner 



In nuclear physics, such forces are called Wigner forces. The 
interaction does not cause any exchange between coordinates of 
the two particles. Another type of interaction is one that inter- 
changes the space coordinates of the two particles in addition to 
multiplication of # by some V(r); for such an in ferae tiou, the 
Schrodinger equation is: 

[(n 2 /M)V- + Ety(r h Z& « u <r 3 ) = V(r)f(i-2, *i, in,*& (146) 

Major ana 



S2 



QUANTITATIVE THEORY OF NUCLEAR FORCES 



Such a force is called a Majorana force. Two other possibilities 
are: (1) the Bartlett force, with interchange of spin coordinates, 
and (2) the Helsenberg force, with interchange of both space and 
spin coordinates. The Schrodinger equations are respectively: 

[(h 3 /'M)¥ 2 + E\^(r u t 2 , ft, a,) = 7(r)^(r 1; r 3 , *% ft) (147) 

Bartlett 

[(hVAf)V 2 + m(r h r 2 , ft, cj) = V(r)#(t2 t r,, <r 2 , ft) (14S) 

ff&senherg 

Effects of Exchange Forces. Exchange forces, with a V(r), are 
central forces and do not cause mixing of I's. However, if a tensor 
force is used instead of V(r) as the multiplying potential, I's are 
mixed and the quadrapole moment of the deuieron may be ex- 
plained as before. It should be pointed out that the tensor force 
does not by itself lead to saturation; this was proved by Volkolf 
(Phys. Rev. 62, 134). 

Majorana Force. The Majorana interaction replaces (r) by 
(— r) in •■]/-. Using the well-known behavior of the wave function 
on such an inversion, the Schrodinger equation (146) may be 
rewritten 

UhVAOV 2 + EU® = (-l) ! 7(r)^(r) (149) 

This is equivalent to having an ordinary potential that changes 
sign according' to whether I is even or odd, and is independent of 
spin. Since the experimental data discussed so far give informa- 
tion on the potential only for 1= % we have as yet no direct evi- 
dence as to wh.et.her the potential is "ordinary" or of the Majorana 
type. Since the potential is attractive for I = 0, it would be 
equally repulsive for I = 1 M the interaction, were totally of the 
Majorana type. 

Bartlett Force. Considering still a system of two particles, the 
spin function is symmetric if the total spin B is 1, and antisym- 
metric if the total spin is 0. Thus, the Schrodinger equation (147) 
for the Bartlett force may be rewritten;; 

[(h a /w 2 + mm = oo^'f-ww $s® 

This is equivalent to an ordinary potential which changes sign 
between S = and £ = 1. Since we know from neutron-proton 
scattering data that both the :i S and *M potentials are attractive, 
the nuclear force cannot be totally of the Bartlett type. 



SATURATION OF NUCLEAR FORCES 



83 



Heisenherg Force. Combining the arguments of the two last 
paragraphs,' the Schrodinger equation (148) may be rewritten for 
the Heisenbcrg force: 

KhViiov 2 + Mum = c - iy +s+v mf(j) (m) 

This is equivalent to an ordinary potential which changes sign 
according to whether l+'$ is even or odd. For example:, the 
effective potential is: 



for 3 5' 

potential + V(r) 



■v(f) 



Sp 

-Y(r) 



+ 70') 



(152) 



The reversal of sign, between 3 $- and ^-states indicates, as for 
flu > Bartlett. force, that the nuclear force cannot he wholly of the 
Heisenherg type. However, the difference between the S S and *£ 
neutron-proton well depths (about 21 and 12 Mev, respectively, 
f or a _ 2.8 X I0~ ri cm) can be explained by assuming that the 
interaction is about 25 per cent Ileisenberg or Bartlett and 75 per 
cent Wigner or Majorana. 

Exchange Forces and Saturation. The Bartlett spin-exchange 
force does not lead to saturation of the binding energy per particle. 
If the nuclear force were of the Bartlett type, heavy nuclei should 
exist with all spins aligned where the number of interacting pairs 
is A (A - l)/2, which leads to binding energy proportional to at 
least the square of A. 

However, the space exchange in the Majorana and the Ileisen- 
berg forces does lead to saturation because of the alternation in 
sign of the potential between odd and even I. For example, assume 
the nuclear force is the Majorana type (we already know it cannot 
bo more than about 25 per cent Heisenherg). Then saturation 
should not be apparent in nuclei up to He 4 , for in He 4 the spatial 
wave function can still be symmetrical in all four particles, without 
violating the Pauli principle. We need only give antiparallel 
spins (antisymmetric spin wave functions) to the two n.iutrons, 
and likewise to the two protons. Thus the Majorana force does uot 
alter the Wigner argument about the short range of the forces 
based on the binding energies of He' 1 and lighter nuclei. 

In the next heavier nucleus— I Ie B or hi 5 — the Pauli principle 
can no longer be satisfied by spin wave functions alone; there- 
fore, the spatial wave function must have at least one node. In 



S-i 



QUANTITATIVE THEORY OF NUCLEAR FORCES 



other words, only four particles can be in an ,s-state, whereas the 
last has to be put in a potato, and will therefore be repelled by 
the other particles. He u and Li 5 should thus be unstable, in 
agreement with experiment. This is a first sign of saturation. 

To investigate saturation in heavy nuclei, one may use the 
same variational method used at the beginning of the present- 
chapter to prove that ordinary forces do not give saturation. 
It i.s satisfactory that this calculation, in the case of the Majorana 
force, does not lead to non-saturation. On the other hand, since 
the variational method gives only a maximum to the true energy, 
it cannot be used to prove that the Majorana force does give 
saturation. Bat Wigncr ha.s given a conclusive argument that 
saturation is achieved with the space-exchange Majorana force 
(Proc. N T at. Acad. Set. 22, 662, .1936). The space-exchange part 
of the Heisenberg force would also cause saturation. 

SPIN AND ISOTOPIC SPIN 

It is often convenient to write exchange forces in a slightly 
different way . Sin c e f o i ■ t w o par tides 

0"i ■ o% = +1 for S = 1 

= -3 for S = 0, (153) 

the Bartlett force between two particles can obviously be written as 

[+v¥&s- i 

-V(t), s = 



W®(X + *i ■ <%) 



(154) 



The spin-exchange part of the Heisenberg force; could be written 
in the same way. 

In order to be able to use a similar notation for the space- 
exchange part of forces, we introduce the concept of the charge 
of a particle as a coordinate, i.e., neutron and proton are regarded 
as different eigenstates of the same particle, called a nudeon. We 
choose the symbol t for this charge coordinate and we define 



My = }4 for the proton 
M T m — ]/i for the neutron 
T = Y % for both 



(1-55) 



SATURATION OF NUCLEAR FORCES 



85 



using + H in analogy with the spin coordinate. We also define; 
the charge functions 



Charge function = y for the proton 
= 5 for the neutron 



(156) 



in analogy with the spin functions a and 8. 

The nucleons must obey Fermi statistics in order to be con- 
sistent- with the ordinary theory (this will become apparent 
shortly, if it is not immediately obvious). Thus the total wave 
function (including the charge function) for two or more particles 



4> = #spafiB&3 'AspmW vWrKeO) 



(157) 



must be antisymmetric with respect to inter change of ail coordi- 
nates of two iiueleons. We therefore look for symmetric and anti- 
symmetric charge functions for two particles. Then; are four of 
these, as given in Table 4. 

TABLE 4 

Twq-Pa:rticle Charge Functions 

Rtspi'i.;- Net 

State Functiou seating Symmetry Charge 

I 7(l)v(2) He 2 symmetric 2e 

II 5(I)5( 2 ) ™ 2 symmetric 

III (1./v / 2)[t(1)5(2) +7(2)5(1)1 H- symmelEe e 

IV (l/v / 2)[7(l)5(2) — 7(2)5(1)1 IT- antisymmetric c 



Again, in analogy to spin, two quantum numbers are defined to 
describe these functions: T to describe symmetry, and M T to 
describe the net charge. These quantities have the values given 
in Table 5. 

TABLE 5 

Quantum Nu-Mfstchs fok Chabge States 



Stale 


T 


M r 


I 


1 


I 


II 


1 


-1 


in 


1 





IV 









W QUANTITATIVE THEORY OF NIT CLEAR FORCES 

T is 1 for symmetric functions, for the antisymmetric function, 
in analogy to spin. M r is the sum of the M r 's for the two nueleons. 

In the literature t is called the "isotopic spin," T is called "the 
total isotopic spin," and M T may be called the "component of r 
in the direction of positive charge." T is analogous to total spin S, 
and Mj to B z . Tor a given T, M T can have the values T, T — 1, 
-■■,-T. 

From Table 4 it is seen that a system containing two neutrons 
or two protons has a symmetric charge function. Since we arc 
assuming nueleons to obey Fermi statistics, the remainder of the 
wave function (147) must be antisymmetric. This implies (cor- 
rectly) Fermi statistics for neutrons and protons, disregarding 
charge as a coordinate. But in a system containing a neutron and 
a proton the charge function can be either symmetric or anti- 
symmetric, and so also can the remainder of the wave function. 
Therefore, the treatment of proton and neutron as two eigens fates 
of the same particle does not in this case introduce any restric- 
tions, consistent with the ordinary theory of statistics-. 

it; is also convenient to introduce an operator t in analogy to 
the cr operator, defined by its effect on the "charge coordinate" 
M T . The eigenvalue of its absolute square is, again in analogy 
with spin: 

|t| 2 - Vf(T + 1) (158) 



Then, just as for spin, in a system of two nucleons 
Ti -To = +1 for T = 1 
-3 for T = 



(159) 



Now the Heisenberg interaction can be written (letting V(r) 
absorb the factor —1) as 



¥>V{r){\ + T] -t.) 



(160) 



To prove this, we note that equation 100 changes sign according 
to whether the charge part of the wave function (equation 157) 
is symmetric or antisymmetric, i.e., according to whether the 
product of space and spin functions is antisymmetric or symmetric, 
which is just what is required according to equations 151 and 152. 
The types of interaction between the two particles discussed so 
far may now be summarized by listing the various types of oper- 



SATURATTON OF NUCLEAR FORCES 



87 



ators, which when multiplied by some V{r) give the interactions 
listed in Table 6. 

Eisenbud and Wigner (Proc. Nat. Acad, Sei. 27, 281) have shown 
that these interactions and their linear combinations are the onlv 



TABLE G 
Types of INTERACTIONS 



Ordinary 

Spin cxdiiinge 

Spaee^spin exulian^o 

Space exchange 

Tensor 

Tenser exchange 



1 

<H 0-2 
Tl ■ T2 

(<T1-0- 2 )(ti-to) 

(<ri-r)(o-2-r) 

(q-l ■ r) (0-2 ■ r) (Tl • Tn) 



Ones possible under certain reasonable invariance requirements, 
namely, excluding interactions depending on total eharge or on 
the momentum. (The in tea 'action (tri H- ov>) • L depends on the 
momentum.) 



QUANTITATIVE THEORY OF EXCHANGE FORCES 

In the last chapter, it was shown that the ground state of the 
deuteron, the neutron-proton scattering, and the quadrupole 
moment, of the deuteron could be obtained quantitatively by 
assuming a neutron-proton interaction of the form 



V 



(even) 



= -(1 - ¥& + US °T ' *z -I- ySrd.Kr) (161) 



with 



Jir) = V'o 


r < a 


J(r) = 


r > a 


g = 0.0715 


V = 13.89 Mev 


7 = 0.775 


a = 2.80 X m~ 13 cm 




— Rarita anil Sch winger. 



The neutron-proton interaction (.161) applies only to states of 
L = 0. The potential for other L is as yet arbitrary. If wc assume 
in particular a fores of the type discussed in this chapter, i.e., 
depending only on the product of the isotopic spins X\ ■ r 2 , the 
potential will depend only on the parity of the state. The poten- 
tial for states of odd parity can only be determined from that for 



88 



QUANTITATIVE THEORY OF NUCLEAR FORCES 



slates of even parity by making some assumption regarding the 
exchange character (or dependence on Tj ■ fa) of the forces. 
Rarita and Sch winger chose; to investigate three potentials which 
were suggested by three types of meson theory (see Chapter XV) : 
I. Symmetric meson theory. 
II. Exchange forces, or charged meson theory. 
III. Ordinary forces, or neutral meson theory. 
These potentials are: 



I. 

II. 

III. 



" = 


— }ifi • t 2 o-i 


0"2 V&yim 


[■' = 


(-1)' 


v 

* even 


V = 




V e veil 



(162) 



where V cvm is given in equation 161. 

For ordinary forces III, the potential in odd states is the same 
as for even. Exchange forces II, on the other hand, have opposite 

sign in odd states. To determine the behavior of the force sug- 

TABLF 7 

PriOFEitTiES ov a Neutron -Pkotcn System 











Is o topic 






Slate 


Parity 


Spin 


s 


Spin T 


<TJ ■ CT2 


T1-T2 


>iS 


oven 







1 


-3 


1 


3 S 


even 


1 







+1 


-3 


!p 


odd 










-3 


-3 


3p 


oild 


1 




1 


-: 1 


+1 



gesled by the symmetric theory I, Table 7 of values of crj ■ ffs 
and ti ■ t 2 has been construe ted for even and odd states of both 
the singlet and the triplet types. 

From equation 162 and Table 7, the symmetric theory (I) gives: 

"f''odd = ~~ /a f'evcii , (M , 

(163) 

Todd — ~~ 3 Y even 

The three types of forces may now be compared with experi- 
ments by computing neutron-proton scattering at high energy. 
The energy chosen by Rarita and Sell winger was 15.3 Mev, for 
which P-wavc scattering begins to be important. The P-wave 
scattering is to be computed with the aid of equations 162 and 163, 



SATURATION OF NUCLEAR FORCES 89 

which give the potentials acting in the P-state. It should be noted 
that in contrast to the usual theory in which a single phase shift 5 1 
is computed for scattering in the P-state, three phase shifts ij , 
i} U and 5j 2 must be computed for scattering by the 3 P -, 3 Pi-, and 
3 P 2 -states, respectively. The reason for this is that the effective 
potential well for each of these three states differs because of the 
presence of the non-central tensor force S&, Tn fact, the operator 
S 12 has definite values (-4 and 2) for the states ;J P and S P X 

TABLE 3 

Well Depths in the Netjtbal Theory 

Effective Well Depth with 
Stale "Ordinary Forces" 

3 P 29.2 Mcv (repulsive) 

3 P i - 3 5 . 4 Me v (attracti ve ) 

3 P 2 - 9.6 Mev (attractive) 

which occur unmixed and must therefore be eigenf unctions of Si 2- 
The 3 P 2 -state has a fairly definite value of S v > (—%}, since at 
15.3 Mev it is only slightly coupled to the 3 P 2 -state. (See Chapter 
XIII for a discussion of how 81$ couples states of different L but 
the same J.) 

Rarita and Schwinger (Phys. Rev. 59,556, 1941), using equation 
161 and the values of S VA just quoted, give the effective well depths 
for the 3 P-statcs in the neutral theory III as shown in Table 8. 

TABLE 9 

Phase Shifts in 3 Po, 3 Pt, and 3 F 2 States 



Tlieoi-y 


11! 


VI 


12 


I 


0.074 


-0.05-1 


-0.017 


II 


0.531 


-0.114 


-0.046 


III 


-0.102 


0.995 


0.073 



The potentials of the charged theory II have opposite sign to the 
tabulated values; those of the symmetric theory I have: opposite 
sign and a, re one-third as large. (See equations 1(32, 163.) The 
phase shifts for each of the three theories, using these well depths, 
are given in Table 9. 

Note that the phase shifts in Table 9 for theory I are small be- 
cause potentials arc used which are only one-third as large as for the- 



90 



QUANTITATIVE THEORY OF NUCLEAR FORCES 



pries II and III. ( S e e eq uati on 1 G3 . ) N o to al so that th e s igns o f the 
phase shifts art; opposite in theories II and III because this is 
also true of their potentials. (See equation 162.) Note further that 
really large phase shifts occur only for strong attractive potentials, 
i.e., tjo in theory 11 and 17! in theory III. 

If the scattering contributions from the 3 P-states are added up 
with the proper statistical weight (2,1 + 1) the total scattering 
for 3 JP-states at 15.3 Mev is found to be: 



1. 

II. 

III. 



<r(8) - X 2 (0.0038 + 0.0045 cos 3 0) 
<*{#) - X- (0.103 - 0.002 cos 2 8) 
<r(8) = X 2 (0,187 + 0.687 cos 2 0) 



(164) 



with 4ttX 2 = 0.082 X 10 -34 cm 3 . 

The scattering is also computed for the (?Bi + ? 'Bi) state. This 
is added to equation 164, taking proper account of interference 
terms with the result that the total triplet scattering in barns 
becomes : 



I. v(8) = 0.080 (0.983 + 0.002 cos fl + 0,051 cos 2 3) 

11. a{0) = 0.746 (0.98G + 0.193 cos 3 + 0.041 cos 2 (?) 

Ill <f(g) = 1.165 (0.857 + 0.849 cos + 0.429 cos 2 0) 



(165) 



The quantities in equation 165 are so normalized that the numbers 
outside the parentheses represent the total cross sections. 

A corresponding calculation for the l P and i S scattering gives: 



1. a(0) = 0.444 (0.939 - 0.438 cos 3 + 0.182 cos 2 6) 

IT. &(§) = 0.424 (0.985 - 0.240 cos 8 + 0.044 cos 3 0) 

111. a(0) = 0.437 (0.955 + 0.498 cos + 0.1.34 cos 2 8) 

where the potentials used in the 1 P state were : 

I. 'V( l P) = -37( 1 .8) = 4-35.7 Mev" 

II. Fit 1 ?) = -Vi'S) = +11.9 Mev 

TIL V ( ' P) = V(\S) = -U.9 Mev 



(166) 



(107) 



Note that the difference between a repulsive force (I and II) and 
an attractive force III is shown by the sign of the teiTn in cos 9 in 






SATURATION OF NUCLEAR FORCES 



01 



I 


0.621 barn 


II 


0. 6G6 barn 


III 


. 983 barn 



equation 106, which represents interference between the l P and 
the l S states. 

The total cross section can be obtained by adding the triplet 
and the singlet scattering in a 3-to-l ratio. The three theories 
give hi fractions of a barn the values shown in Table 10. 

The total cross section should not be used by itself to mate a 
definite decision between the three theories since it is influenced 

TABLE 10 

Theoretical Xijutr on' -Proton ScATTEHora at 15.3 Mev 

Theory Total Cross Section Angular Distribution 

1 - 0.080 cos + 0.077 coir 8 
1 + . 126 cos -f . 042 cos ! 8 
1 -h . 032 cos + . 4r>7 cos 2 8 

by the range and the shape chosen for the interaction potential. 
On the other hand, the angular distribution is good evidence for 
the existence or non-existence of strong P-scattering, and also 
gives the sign of that scattering— thus providing direct informa- 
tion about the exchange nature of the neutron-proton force. 

For comparison with experiment, we may note from Table 10 
that at 15.3 Mev, theory I gives a weak backward maximum, 
theory II a weak forward maximum, and theory III a strong 
forward maximum. 

EXPERIMENTS ON NEUTRON-PROTON SCATTERING 

Total cross sections can bo obtained by measuring the absorption 
of neutrons in paraffin and correcting for the presence of carbon. 
Angular distributions have been measured by Amaldi and others 
(N atu rwissen seh af ten 30, 582, 1942; also Ricerca scientifica 
1942), using the recoil protons projected from a paraffin foil. 
The proton directions are determined by the use of a coincidence- 
counter "telescope." Proton ranges, hence energies, are deter- 
mined by the simultaneous use of absorbing foils. 

In the center- of-m ass system, conservation of momentum re- 
quires that the neutron and proton leave each other in opposite 
directions — i.e., at angles 8 and 180° — 8 to the incident neutron, 
respectively. In the laboratory system, the two particles leave 
at right angles to each other, and the angle between proton and 
incident neutron is 90° — 8/2. 



<Xi 



QUANTITATIVE THEORY OF NUCLEAR FORCES 



Amaldi found that the number of protons projected forward 
was small, corresponding to weak neutron scattering in the back- 
ward direction, 6 = ISO 3 . This is in agreement with ordinary 
forces III and in contradiction to exchange and symmetric theories 
II and I. Amaldi measured R = o-(180°)/V(90 D ), the angles 
being the neutron scattering in the center-of-mass system. His 
results are given in Table 11 together with their quoted accuracy. 

TABLE 11 

High-Energy Xeutu m -Proton SciSTBKnJS (Amm.iit) 
It 



E (in Mev) 
12.5 
13.3 

14.0 



= aCLScryM'JO ) 

0.71 ± 0.04 
0.53 ±0.03 
0.52 ±0.03 



The values of 7? at 15.3 Mev computed from the cross-section 
formulas in Table 10 give for the three theories: 



I. R - 1.157 



II. R = 8.916 



III. B = 0.525 (168) 



On the other hand. Champion and Powell (extension of experi- 
ments reported in Proe. Hoy. Soe. 183, 64, 1944). using neutrons 
of similar energy and using photographic techniques, find that the 

scattering is practically isotropic. However, their experimental 
data have less good statistics and greater correction factors than 

Amaldi's.* 

More definite evidence contradicting Amaldi's results comes from 
measurements of the proton-proton scattering at energies of 14.5 
Mev by It. R. Wilson and collaborators (Phys. Rev. 1947). Al- 
though these experiments are preliminary, they indicate a slight 
repulsion in the P-state. They might be reconcilable with ex- 
change forces or with zero forces in the /'-state, but they appear 
to fit best to a force of the cr, -0-2 type and they certainly contradict 
an ordinary force such as would be required by Amaldi's experi- 
ments. There is, of course, the logical possibility that neutron- 
proton and proton-proton scattering are different, but. in any ease 
the present state of this subject is inconclusive and more accurate 
measurements are urgently needed. 

* Laughlin .and Kruger (Phys. Rev. 71, 736, 1947) also tind isotropic dis- 
tribution (at 12-13 Mev).— Note added in pro>>f. 



SATURATION OF NUCLEAR FORCES 



03 



If Amaldi's results are correct they imply that the forces in the 
P-state art; attractive, and they support the theory of ordinary 
forces III. Unfortunately, this result cannot be easily reconciled 
with the saturation property of nuclear forces.* 

* Experiments carried out with, the 184-irich. cyclotron of the University of 
California at the end of 1946 demonstrate definitely the exchange nature of 
the forces between neutron and proton. It was shown in these experiments 
that a neutron of about 100 Mev will produce protons mostly in the forward 
direction and with energies neatly equal to J 00 Mev. This had been predicted 
by Wick for high energy collisions between neutrons and protons. If the 
forces were ordinary forces the proton would in general receive an energy 
of the order of the depth of the nuclear potential well, i.e., about 10 Mev. On 
the other hand, if the interaction is of the exchange type, then neutron and 
proton will change roles: the neutron will re Lain an energy of the order of 
10 Mev and the proton will take almost the entire energy. When this note 
was written it had not been established whether the forces are of the pure 
exchange type or of the type corresponding to the symmetrical meson theory.— 
Note added in proof. 



XV. SKETCH OF THE MESON THEORY 
OF NUCLEAR FORCES 



This theory is presented although it Iras so far not given any 
results in quantitative agreement with empirical facts on nuclear 
forces. However, it may give a valuable point of view. 

The Coulomb force between two charged particles can Ik; ex- 
plained in terms of the interaction of these particles with the elec- 
tromagnetic field. Similarly, the force acting between two nucleons 
might fee described by a meson held surrounding the first particle 
which acts on the second. 

Moving charges produce a radiation field which can be quantized 
and described in terms of photons. The "quanta'' surrounding 
a nuclear particle arc called mesons. Yukawa, in initiating the 
meson theory (Proe. Phy si co-Math. Soc. Japan 17, 48, 1935), 
suggested that if the mesons are given a finite rest mass m. the 
range of forces arising from the meson field will be h/mc, the 
Compton wave length for the meson. If the range of nuclear 
forces is assumed to be 2,8 X 10 _1B em, the meson rest mass 
should be about 140 electron masses. Particles with about this 
rest mass were discovered in cosmic rays two years later. In the 
meantime, Erode- and Fretter have determined the rest mass to 
be 202 ± 10 electron masses, giving a range of 2 X 10' I,! cm. 

To determine the nature of the meson field and the correspond- 
ing nuclear forces, an equation analogous to V- $ = — 4irp must be 
written for the static part of the electromagnetic field. A rela- 
tivist! c equation suited for particles with no spin and a finite rest; 
mass m is the Klein- Gordon equation : 

?V + (l/nV)[(E - TO 2 - (mc 2 f]4t = -Ittp (169) 

with 

E = ih(d/di) (169a) 

where p in this case is proportional to the density of nucleons. In 
free space, V = 0. For a static meson field, according to equation 
169a, we must put E = 0. Furthermore;, if there is one point- 
nucleon at the origin, the Klein-Gordon equation, becomes 

W% - {mefMH. = 4ir E / 1 5(r) (170) 

94 



MESON THEORY OF NUCLEAR FORCES 



95 



where o represents the Dirae ^function , and g L is a constant 
replacing the electronic charge in electrodynamics. 

The solution of this equation is 

$ = -(d/r) exp [-(jn.c/h)r\ (171) 

and the potential acting on a second nucleon is given by: 

V = tyfr 0"2) 

where ffi and g 2 are the effective nucleonic "charges" or coupling 
constants. 

The Yukawa scalar meson theory just described produces the 
required range for nuclear forces. Since in this theory the nuclear 
particle does not change its nature (i.e., charge) we find that 
according to tire theory the neutron-neutron, neutron -pro ton and 
proton-proton forces are all equal. However, the theory does not 
explain the spin-dependence of nuclear forces. Furthermore, the 
forces are all "ordinary," whereas exchange; forces were found to 
be necessary to explain the saturation of nuclear forces. 

Since tire mesons discovered in cosmic rays were; all charged 
either positively or negatively, a theory of charged mesons was 
developed. According to this theory, the following reactions can 
take place: p^# + A + m N ^ p + ^- (]73) 

Thus protons and neutrons can transform into each other by the 
emission or absorption of positive or negative mesons. The 
interaction between two particles, 1 and 2. can take place, for 
instauee, by the following scheme: 

Pi -* Ai + ^ -Y 2 + M + - > i\ (174) 

It is clear that such an interaction can only occur between a 
proton and a neutron, not between two like particles. This is in 
contradiction to experimental evidence and rules out the charged 
meson theory, at least in tin; case of weak coupling between 
nucleons and meson field (small value of g). Further, the charges 
of particles J and 2 are exchanged in the process of emission and 
reabsorption of the meson: therefore; this meson theory leads to a 
force of the charge exchange; or Heisenberg type. This, while giv- 
ing saturation, is in contradiction with experiment (Chapter XIV). 
To explain the neutron-neutron and proton-proton forces which 
are missing in the charged theory, a symmetric scalar meson theory 
was developed, containing neutral, positive and negative mesons 
described by three functions #i, fa, and ifo. To get spin-dependent 



90 



QUANTITATIVE THEORY OF NUCLEAR FORCES 



V = g i 



+ °V°"2 



(175) 



nuclear forces, the meson field must further depend on the spin of 
the nucleon which generates the field. This is achieved by intro- 
ducing into the Tlamiltonian of nucleon plus meson field, an inter- 
action energy containing the factor cr ■ grad $ where u is the 
nucleon spin. In this case $ must be a "pseudoscalar" since <r is 
an axial and grad a polar vector. (A pseudosealar changes sign 
-when the sign of the time is reversed, or on inversion of the Spatial 
coordinates; under Lorentz transformations, it is invariant.) 

Solution of the symmetric pseudosealar meson field equation 
led to an interaction energy between two nucleons of the form 

i r /3 :v A 

-Tl-T 3 s ia i-.|-.-+-i 
3 _ \r r r / 

where i± = mr;/h. 

The term in o-j - cr 2 provides the spin dependence of nuclear 
forces, and the tensor force S l2 explains the existence and sign of 
the quadrupole moment. All these features are in qualitative 
agreement with experiment, as shown in the preeeding chapters. 
Unfortunately, the high singularity of V at r = makes it impos- 
sible to solve; the Schrodinger equation. 

Two ways of saving the situation have been suggested: (1) to 
cut off the interaction at some finite radius r u , i.e., to give the 
neutrons and the protons a finite size, or (2) to mix two meson 
theories in such a, way as to eliminate the undesirable singularity. 

Tie assumption of finite sources (1) unfortunately caimot be 
formulated in a relativistic invariant way. Furthermore, use of 
the rigorous relativistic interaction between nucleon and meson 
field leads to the reappearance of terms in \/r and l/r s in the 
"mixed" theories, in higher approximations. Therefore there are at 
present no trustworthy results of the meson theory of nuclear forces. 

It should be noted that many of the statements made about the 
spin and charge dependence of the nuclear forges have to be modi- 
fied if the coupling between nucleon and meson field is strong, i.e., 
if many mesons are emitted simultaneously. The coupling con- 
stant for an electromagnetic field is (?/lic = MV 7j "■ SB6&EI value, 
whereas that for the meson field cf/hc ca 14 or % is considerably 
larger. The divergence of the interaction at sma.ll distances 
makes the interaction effectively even stronger. For this reason, 
much effort has been spent to treat the strong coupling problem 
in meson theory, but so far no results have been obtained which 
throw light on the problem of nuclear forces. 



C TOPICS NOT RELATED TO NUCLEAR 
FORCES 



XVI. BETA DISINTEGRATION 

In Chapter VI, experimental evidence was given for the hy- 
pothesis of the production of neutrinos of rest mass and spin }/ 2 
in 0-decay processes. This assumption made possible the conserva- 
tion of energv and spin. The first detailed theory of the process 
was given by Fermi (Zeitschrift fur Fhysik 88, 161, 1934). A 
modification which seemed necessary but was later abandoned 
was the work of Konopinski and Uhlenbeck (Pbys. Rev. 48, 7, 
1935). A summary is given by .Konopinski (Rev. Modern Phys. 
15,209, 194.3). 

Fermi introduced a new interaction between the nucleon and 
the two light particles, electron and neutrino. His interaction 
was chosen in analogy with the interaction between charges and 
electromagnetic field in quantum electrodynamics. (This analogy 
was also used in the last chapter in connection with the meson 
theory of nuclear forces.) The heavy particles are to act as sources 
and sinks of the light particles. 

If the Hamil Ionian of the interaction between the proton, 
neutron, and electron -neutrino fields is H ; then the number of 
transition processes per unit time is 



(2^/h}|J> fitI .*^,,^ 3 • pQB) 



(176) 



where p{S). — the number of final states of the system per unit 
energy interval 
=s initial state of the system 
= u^. = initial state of the nucleon. 
= Mfin. ' &fee. ' <?n. = Spal state of the system 
= (final state of nucleon) ■ (final state of electron) 
■ (final state of neutrino) . 



to 



Fermi's assumption for // was essentially 

Jfe* Hto„ (It - gfu m * fee* <^.ifin. & (177) 

(neglecting relativistic corrections which are important only if the 
heavy particle has high velocity) where fee. arul S8t. arc to bt; 

97 



98 TOPICS NOT RELATED TO NUCLEAR FORCES 

evaluated at the position of the nucleoli, and therefore the integral 
is over the coordinates of the nucleoli alone. This is similar to 
the case of electrons and light: a charge can only interact with a 
light quantum, when they are at the same place. The constant § 
which determines the strength of the interaction must be found 
from experiment. It has the dimensions erg ■ cm 3 , since 1^1^. 
and <p n . are Lo he normalized per unit volume. 

Note that we use ^i^,*, bat p a , (without a star). This corre- 
sponds to the emission of an election but the absorption of a 
neutrino. However, this absorbed neutrino can be taken from a 
state of negative energy which corresponds to the emission of an 
'■'antineutrino." Owing to the absence of charge and magnetic 
moment, an antineutrino is equivalent to a neutrino. The formu- 
lation (177) is therefore equivalent to the emission of an electron 
and a neutrino, and it is a mathematical convenience to have 
formally one particle absorbed and one created. The positron 
emission would be described by #ejes.*k* 

Since the neutrino has very little interaction with anything, its 
wave function may be taken as a piano wave. If p n . is the; mo- 
mentum of the emitted antinouirino, then — p n . is that of the 
absorbed neutrino of negative energy, and 



Pa. = T ! -exp (-i p ru - r/fi) 



(178) 



where V is the volume of a box in which the wave function is 
normalized. The factor V~ 1 ' 2 may be omitted if a unit volume is 
used for the normalization. $&«. should be a Coulomb wave func- 
tion; but if Z the charge number is small, the Coulomb energy of 
the electron can be neglected in comparison with its kinetic energy 
and. a plane wave can be used for the electron wave function. 
The number of final states per unit energy is 



P (m = 



(Volume element of mo- (Volume element of mo- 
mentum space of electron) men turn space of neutrino) 
(Volume of phase space per (Volume of phase space per 
electron energy state) " neutrino energy state) XdE 



= (Jeta 2 #ete $»ri«J IP". 3 # B . doi n )/(2rhf d.E n , 
where do> elvi .da) n _ are elements of Solid angle. 



(179) 



BETA DISINTEGRATION 



The result for the transition probability of an electron into 
dS d[iC . and solid single i&f&w (integration over all directions of the 
neutrino has been carried out) is 



G 2 mc? I r 
§^~h~P^ 



exp 



-i(pn.+P C ]o .)- 



2 , m 

rfre( e 3 -l)^ U -0 2 rfe- 
(180) 



with G = (f//mc 2 )(h/mc) % e = E^Jmtr, V r - I. = p^/jrac, 
eg = E^, llihM Jmc 2 . A plane wave has been substituted for the 
electron wave function. 

Just as in the theory of atomic transitions, there will be selection 
rules for /3-deeay processes. If p„ h: ... and p n _ are both of the order 
of magnitude mc. as is usually the case, the exponent (p„ -f- p (!let .,) 
■ r/h will be of the order of magnitude; 



4 X 10"" cm 



h/mc 3.85 X I0~ n cm 



1 

100 



(181) 

(R = nuclear radius; medium-weight nuclei have been chosen.) 
Thus, exp [i(p n . + p ek .c.) ■ r/h] will be nearly 1, and the matrix 
element in equation 180 reduces to M = J&fia.**^. dr ; i.e., to an 
expression depending only on the state of the nucleon before and 
after the transition. M is determined by the nuclear wave func- 
tions. In parti cular, the orthogonality of the nuclear wave func- 
tions for states of different angular momentum I gives the selection 

ni,e: M fi implies AT = (182) 

Such transitions are called allowed. Transitions for which M = 
are called forbidden; in this case the exponential in equation 180 
must be expanded in a power series; the order of the forbidden 
transition is the number of the first term in this power series which 
gives a n on -vanishing result for the matrix element. Because of 
the estimate (181), the probabilities should decrease by a factor 
of about 10 4 with each order. 

ALLOWED TRANSITIONS 

The only depend ence of the allowed transition probability on 
the electron energy is through the volume clement in momentum 
space. The energy spectrum of electrons is therefore 

KB de ~ el/e s - Ifo - tf d* (183) 



100 



TOPICS NOT BELATED TO NUCLEAR FORCES 



Since eo is unknown, the experiments have to yield a value of t#, 
while giving a check on the theoretical spectrum.. This is easily 
done by making a "Kurie plot." In this plot, the quantity 



F(e) = V;V(f)/£(i 2 - 1) H 



(184) 



According to 
hould vie Id a 



F<t) 



(as observed) is plotted against the energy & 

equation 183, F(e) <--■ e () — e; therefore the plot 

straight line which cuts the e-axis at eg. 

The only nucleus which checks this proportionality exactly is 

In 114 , measured by Lawson and Cork (Phys. Rev. 57, 982, 19-10). 

Hero to = 1.99 M.ev (which is 
high enough to make the ex- 
periments on the /3-rays fairly 
easy) and the lifetime is 72 
seconds. Luckily this short- 
lifetime /3-dccay follows a 50- 
day-lifetime 7-decay (isomeric 
transition; see Chapter IV). 

There are experimental dif- 
ficulties in flic measurement 
of the energy spec tin of most 




Thick 
target 



0.615 Mev E 

Fig. 13. Kurie plot of the positron 
spectrum from Cu . 



other /^radioactive nuclei which result from either the low energy 
of the electrons or the short lifetimes. Cu M measured by A. W. 
Tyler (Phys. Rev. 56, 125, 1.939) emits both positrons and elec- 
trons. The positron spectrum was measured both for thick target 
and thin target (thick and thin relative to the electron range). 
The Kurie plots are shown in Pig. 13. It is not known whether 
the portion AB of the thin target curve is spurious or results 
from another decay process (to an excited state of JMi* 54 ) with a 
very low energy limit. 

The thick target curve is typical of the experimental evidence 
which lead Konopinski and L'hlenbeck to introduce then alterna- 
tive theory (Phys. Rev. 48, 7, .1935). They proposed using the 
time derivative of the neutrino wave function &&J&i in the transi- 
tion probability instead of ip. Since &/>/ dl o-< (sq — e)^ this led to 

spectrum 

eVe 2 - l(«u - <0 4 de 



A 7 k-u de>~^ e - * 



(185) 



thereby moving the maximum of the spectrum to lower electron 



BETA DISINTEGRATION 



101 



energies. To make a Kurie plot of this, the fourth root must be 
used in equation 1S--1 instead of the second. Many of the experi- 
mental data on thick targets then give straight lines but very high 
values of €<>. Later experiments us ing thin targets showed that 
the Kurie plots according to the Konopinski-TJhlenbeck theory 
dropped off, as shown in Pig. 11, which demonstrated that the 
straiglrt-line portion was accidental. Also, when the mass differ- 
ences of nuclei became better known, the values of <e given by 
the Konopinski-Uhlenbeck theory were shown to be much too 
Mgh in all cases but that those given by the Fermi theory agreed 
with the measured mass difference. 

A' 13 measured by Kikuchi et al. (Proc. Physi co-Math. Soc. 
Japan, 21, 52, 1939); Lyman (Phys. Rev. 55, 1123, 1939); and 



VV--V 




Fir.. 11. Typical Kurie plot of the Konopinski- Ohlenbcck theory. 

Townsend (Proc. Roy. Soc. A177, 357, 1941), is one case in which 
the use of very thin targets still did not gore a Fermi distribution. 
To account for such spectra it is usually assumed that several 
decay processes an; taking place simultaneously, leading to various 
energy levels of the residual nucleus. With N 13 — » C li + § 
this is confirmed by the observation of a y-ra,y of about 280 kev 
by Richardson (Phys. Rev. 55, 609, 1939). This y-ray is attributed 
to the transition of the residual nucleus C 13 from its excited to 
the ground state. Unfortunately, various experimenters disagree 
on the relative intensities of the 7-rays and of the two compo- 
nents of the /3-spectrum, and on the value of the upper limit of its 
lower-energy component. 

Coulomb Field. In expression 183 for the electron energy 
spectrum no account has been taken of the Coulomb field. The 
correct spectrum has a greater electron density at lo w ener gies. 
There is no zero for e = 1 because the factor V« -l~s 
(velocity) in the density of states Is canceled by a 1/v in the charge 



■ I 



102 



TOPICS NOT BELATED TO XL" CLEAR FORCES 



density of electrons at the nu ele us. The result i ng electron spectrum 

is showa m Fig. 15. 

Far positrons, fewer of low energy should be expected than the 

number given by expression 183 because of the repulsion of the 

positrons in the Coulomb field: 
The Coulomb wave function of 
the electron in expression 177 
has a facto c cxp ( — 2-irZcr ,/hy) , 
which lowers the transition 
probability considerably for low 
velocities. 

There are some disturbing 
measurements by Backus (Rhys. 
.Rev. 68, 59, 1915) on the ratio 
., in the Cu fil ^-transitions: 




Fig. 15. Energy distribution of 
jS-mys with Coulomb field. 

of positrons to electrons, N+/N. 



Cu 64 - 



f \m m + ,3- 

[Zn^ + f 



a calculable way at 



N+/N"_ should be smallest and behave in 
low energies; the experimental values were compared with the 
theoretical prediction but the value of N_j_/N_ was found to be 
ten times greater than predicted. These measurements should be 
repeated. The disagreement can hardly be attributed to a failure 
of 0-ray theory because the ratio of positron emission to /C-electron 
capture was found to be in exact agreement with theory (Scherrer 
et ah, Phys. Rev. 68, 57, 1.9:1:5}, and this ratio Involves parts of 
the theory very similar to those in Backus' experiment, 

LIFETIMES IN ALLOWED TRANSITIONS 

The total transition probability, or reciprocal of the lifetime, 
for /3-ray emission is found by integrating over the energy distribu- 
tion (equation 180) to Ire 



1/V b (G 2 /2tv*) • (mcVh)|ilir/<-(^) 



(186) 



G is a dimonsionless constant describing the strength of the inter- 
action between electron-neutrino and the heavy particles: M is 
the matrix element for the transition : 



M = fu nn *(T)u in _(r)e- i( ^+*< ] -* /h dT 



(187) 



! 



BETA DISINTEGRATION 
F(e ) is the integral of the distribution in energy 



f(& = £*W^ - *& - *f^ 



103 



(188) 



where e is the total energy available for neutrino and electron, 
including rest mass, in units of the electron rest energy. W (*$) 
varies rapidly with e (1 , being approximately equal to ( ! /SO) e s for 
e » 1 aurl to 0.216(fF — l) y ' for sg nearly unity. Thus r de- 
creases rapidly with increasing e , but not as fast as in the case of 
a-decay, where the transition probability is proportional to an 
exponential of the energy. In Chapter II it was pointed out that 
in natural a-deeay a factor of 2 in energy is equivalent to a factor 
of 10 -20 in lifetime. 

The matrix element M is in general not known because we have 
very scant knowledge of nuclea.r wave functions. Even if we know 
that the transition is allowed, we can in general say only that \M\ 
is between zero and one. 

However, in some cases the value of M can be guessed to some- 
what better than order of magnitude. For allowed transitions 
(A/ = 0) , we have 

M «/%*.%, 4r (189) 

M will be near unity when the wave functions u fill . and u- m , are 
nearly alike. Such is the ease for ,6*-transition between mirror 
nuclei (Chapter II) (for which also the selection rule Al — is 
likely to be fulfilled). Three examples of allowed transitions in 
mirror nuclei are given in Table 12. The product iFfa) is remark - 

TABLE 12 

Allowed Transitions in Mihhob Nuclei 



Reaction 


t = half-life 


eo 


tFUn) 


H 3 -a He 3 + (?~ + " 


10* m 


1.03 


1400 


C u -* B" +fT + v 


1200 sec 


2.S6 


3500 


So* 1 -» Ca 4! + D+ + v 


. <J sec 


10. OS 


2500 



Source: Konopinski, Rev. Modern Phys. 15, 209. 

ably constant, confirming the theory underlying equation 186. 
This constancy exists in spite of t varying by a. factor as large as 
10 s . Furthermore, it, is reasonable that, tF is somewhat smaller 
for the first situation than for the other two, for in a nucleus 



104 



TOPICS NOT RELATED TO NUCLEAR FORCES 



containing only three particles we would expect u fm , and i%, to 
be more nearly alike than in the heavier nuclei; so that | M \ would 
be closer to unity in the light nucleus. 

It is interesting to note that the Konopinski-TJhlenbeek theory 
of /3-decay predicts variation by a factor of JO 5 between the prod- 
ucts i-F for the various reactions in Table 12. 

For nuclei of intermediate mass, the Coulomb repulsion already 
introduces considerable asymmetry between the numbers of pro- 

TABLE 13 
Allowed Transitions in Intilkmediately Heavy Nuclei 



Reaction 
g?5 _> C1 3S + ?- + v 
Cu fjl -> Zn M + $~ + v 

Cu M -> Ni i4 -T- 3+ + v 
In 117 -h» Sn 1 *' 4 



+ ' 



19,000 
66,000 

22,000 
140,000 



tons and neutrons (there are no more mirror nuclei), and pre- 
sumably even greater' differences, between neutron and proton 
wave functions in the nucleus. Thus, even for allowed transitions, 
smaller matrix elements are expected for intermediately heavy 
nuclei than for light, mirror nuclei. T his is borne put by the data 
in Table 13. 

In the heavy, naturally radioactive nuclei the matrix elements 



TABLE 14 
Allowed TsajtskSOKS in Naturally Radioactive Ndclf.i 



Emitter 


tF(ta) 


RaB 


50,000 


UX S 


270,000 



are in general still smaller. This is borne out by the data in 
Table 14. 

Assuming \m\ ~ 1 for the lightest., mirror nuclei, can be cal- 
culated from Ft. The result is 



G « 10" 



(190) 



This corresponds to g ~ 10 ' ls erg • cni a . The smallness of this 
coupling between electron-neutrino and the heavy particle is 



BETA DISINTEGRATION 



105 



what makes (3-decay take place so slowly compared to other 
nuclear reactions, except some a -radiation. It is safe to say that 
ji-rayti are not emitted during nuclear collision*, but only at com- 
paratively long times afterwards. For example, the lifetime of 
protons in the sun due to the reaction 



II + H -» D + r + v 



(191) 



is about 10 11 years, even with a density of about 100 and a tem- 
perature of 2 X .10 T degrees C. (See Bethe and Critchfield, Phys. 
Rev. 54, 248.) Even so, this reaction presents about the best 
opportunity for /3-decay during a collision. The long lifetime of 
the proton in the sun indicates an extremely low probability of 
/'i-decay per collision. 

The most fundamental (3-decay is that of the neutron 



n —> II + j3 + v 



(192) 



The matrix element for this reaction should be exactly unity, as 
the wave function for a single proton ought to be the same as that 
of a single neutron. Measuring the lifetime of this reaction 
should give an exa.ct value of G. However, this reaction is hard 
to observe as the neutrons are removed much more rapidly by 
other means (capture, diffusion) than by the above reaction. 
Using the value of (J found above, the half-life for the reaction 
(192) should be about 15 minutes. There is hope of making the 
measurement with the large neutron fluxes now available in piles. 



LIFETIMES IN FORBIDDEN TRANSITIONS 

The second term in the Taylor expansion of the exponential 
in the matrix element (1ST) will give a non-vanishing integral when 
A/" = zhT. which transition was forbidden in the first approxima- 
tion. Similarly, A/ = ±2 transitions become possible with the 
third term in the expansion, and so on. For t = 2, the argument 
of the exponential averages about 1/100 over the range of the 
heavy par tick; wave function, so that \M{AI = =L.L)'| 2 might be 
expected to be about JO" 1 times \M(&I = Q)| 3 . Actually, the true 
wave function for an electron in the Coulomb field varies faster 
than the plain; wave approximation used in equation 187, and the 
factor 10" 1 becomes about 10~" for medium and heavy nuclei. 
This correction does not help the higher forbidden transitions SO 



106 



TOPICS NOT RELATED TO NUCLEAR FORCES 



much as the first. Higher ^ makes all forbidden transitions more 
probable. Table 15 quotes experimental data from Konopinski 
for forbidden transitions in light nuclei. 

TABLE 15 

IIaIjF-Ltves in FoRBIBOEN Transitions 



Emitter t = Half-life m 

First Forbidden, Transitions 



&fa) 



Li s 


0.9 sco 24.5 


2.S X 10 5 


Ne 23 


40 see 8 


10 r ' 




Second Forbidden Transitions 




pn 


1.2 X 10 B see 4.37 
Higher Forbidden Transitions 


8.6 X I0 7 


B<; m 


KV^SCri 2.1 


10 H 


K « 


5.10 16 sec 2.4 


10 11 



Source': Konopinski, Rev. Modem Phys. 15, 200. 

GAMOW-TELLER SELECTION RULES 

Tli ere is good evidence that the selection rule AT — for allowed 
transitions is not generally adhered to. One example is the K- 
eapture reaction ^ + K ^ L . 7 + p (m) 

Li 7 is produced both in its ground state and in an excited state 
about 440 kev above the ground state. The experimental ratio of 
number of transitions to the ground state to number of transitions 
to the excited state is about 1.0 to .1. This is about equal to the 
calculated ratio, using equation 186 and assuming \M\ equal for 
the two cases. From this and the absolute lifetime it may be 
concluded that both transitions are allowed. However, we do 
not expect both states of Li 7 to have the same value for I. The 
best assumption is that the two states form a /'-doublet, with 
I = Yi and I = % for excited and ground states, respectively. 
Thus AT can certainly not be zero for both transitions. 
Another example is the reaction: 

He* -» Li 6 + /T + v (194) 

Li a can be thought of as an ct-particle plus a deuteron. The 
.a-particle has / = 0, and the deuteron has I =» 1. We expect, 



BETA DISINTEGRATION 



HIT 



therefore, that Li a has I = 1, in agreement with experiment. In 

the same picture. Tie 6 is an a-partiele plus two neutrons. In the 
"ground state," the double neutron should have spin zero (ef. 
Chapter XII), so that the same argument gives 7 = for He*. 
An additional argument for this is that all nuclei containing even 
numbers of neutrons and protons have zero spin as far as they have 
been investigated. Thus AI = I, and the transition is forbidden. 
But the experimental lifetime of the reaction shows that it is 
"allowed," There are similar situations in the /3-decay of C i0 , 
F 18 , and Na 3U . 

So it seems that there can be allowed transitions with A 7" = 1. 

Gamow and Teller first allowed how this can come about. They 
said that in considering possible interactions, one ought to include 
all relativistically invariant combinations of the four wave func- 
tions, u- m ., tifijj., &jfeM and 6,... For two wave functions, let us say 
\f/ and 4>, there are five combinations which are co variant under 
Lorentz transformations : 

1. Scalar: ^* j3 <i> (Fermi theory). 

2. Polar four vector, with components: ^* 6, f* a </>. 

3. Tensor: $*■&&■$, V P a <t>- 

4. Axial vector: $* cr <*>, #* 7s *. 

5. Pseudoscalar : $* ,8 y 5 <j>. 

where fi, a, and y 5 are Dirac operators and a is the usual spin 
operator. (For details, see Konopinski's article.) To obtain a 
relativisficaily invariant interaction, the corresponding combina- 
tions of the wave functions of the light and of the heavy particles 
must be multiplied; for example, the tensor combination of the 
light particle wave functions with the tensor combination of the 
wave functions u m , and i( fin . of the heavy particles. In this case 
the Hainiltonian becomes: 



V (tensor) = (f*j3<r4) ■ (u&^&TMia) 



(195) 



(The transition is still treated as though an antineutrino is 
emitted.) Since the heavy particles are non-relativistic, the Dirac 
operator ,3 for them is equivalent to unity; therefore, the net effect 
of equation 195 is to place the operator cr between the heavy 
particle wave functions fe. and « fln- . Therefore, the matrix 
element for allowed transitions is now J ■'u-fi 11 .*<nii il , dr, and this 
may be different from zero if the total sphi I changes by one unit, 



mmm 



10S 



TOPICS NOT RELATED TO NUCLEAR FORCES 



or by zero, in the transition. Thus M = ±1,0 can be "allowed" 
for the tensor interaction. 

The axial vector interaetion gives the same selection rule as the 
tensor, 

Al = 0, ±1 (106) 

From the experimental data it seems that these Gamow-Teller 
selection rules are correct. For instance, they explain the results 
for He 6 , C 10 , F 18 , and Na~. However, the reaction 

Be in -» B 10 + &~ + v (197) 

differs from (194) only by the addition of an at- particle, so that 
&I = 1 may again be expected for this reaction. But experiment 
shows that this is forbidden. The same is true for the reaction 



C 1 



n m + £-- + v 



(198) 



which differs from (191) by two a-partieles. Thus the Gamow- 
Teller selection rides, while;, explaining more than the Fermi rules, 
still are in contradiction with many of fetes data. 

K-capiure. The theory for A'-capture lias been worked out, 
and is in good agreement with experiment. Scherrer el al. (Phys. 
Rev. 68, 57) have measured the ratio of Z -capture processes to 
positron-emission processes for Cd 107 (or loy ?), with the result: 
320 =fc 20. The Fermi theory predicts 340. (The Konopinski- 
Uhlenbeck theory gives 20,000, and is conclusively ruled out.) 






XVII. THE COMPOUND NUCLEUS 

In tins chapter, we are no longer concerned with the determina- 
tion of fundamental nuclear forces, but with the more practical 
problem of predicting cross sections for nuclear reactions, par- 
ticularly those involving heavier nuclei the quantum states of 
which are not known precisely. On the other hand, the presence 
of many nuclear particles will make statistical methods practical, 
and these are used in the theory of the compound nucleus. 

The concept of the compound nucleus was initiated by Bohr m 
1935. In order to get a clear picture of this concept, we shall 
examine the difference between nuclear collisions and atomic 

collisions. 

For collisions between an. atom and a particle of high or moderate 
energy, the Born approximation is valid because the incident 
particle passes right through the atom practically undisturbed. 
Slight deflections, inelastic collisions, and emission of radiation 
are progressively less likely processes. The reason that particles 
are likely to pass right through is that the atom is a loosely bound 
structure. Another way of saying this is that the interaction of 
atomic electrons with, say an incident electron of several thousand 
volts, is much smaller than the incident energy- -which is precisely 
the condition for validity of Bom's approximation. 

Nuclear interactions, on the other hand, are of the order of 20 
Mev, which is much greater than the kinetic energy of the incident 
particle normally used, i.e., several Mev or less. This is precisely 
the opposite of the conditions required for Born's approximation. 
Here, the interaction energy is more important than the kinetic 

energy. 

Another difference: An electron striking an atom can be re- 
garded as interacting with the average "Hartree" field of the atom. 
This approximation is valid because the interaction with a single 
electron is much smaller than the average interaction with all the 
electrons. On the; other hand, the short range and the saturation 
character of nuclear forces require that, nucleons interact only 
with a small number of neighbors. Tims individual interactions 

109 



11.0 



TOPICS NOT .RELATED TO "NUCLEAR FORCES 



will be of the same order of importance as the average total inter- 
action — and it will not be permissible to replace tire nucleus by an 
average field. 

The Bohr picture takes advantage of these large interactions 
and describes them in terms of a compound nucleus. Tire theory 
makes the following statements; 

1. Any 'particle which kits the nucleus is caught. A new nucleus 
is formed called the compound nucleus. The reason for this is 
that an incident particle will interact with one or two nucleons, 
transferring much of its energy to them and thus to the nucleus, 
before penetrating it appreciably. Then it may no longer have 
sufficient, kinetic energy to escape the attractive nuclear forces, 
and is therefore caught. 

2. The compound nucleus is long-lived compared to the natural 

nuclear time. (This is the time for a neutron to cross the nucleus — ■ 

cm 
say 10 ~ r2 cm/ 10" -' - =* 10 ~ 31 second.) The reason for this is 
sec 

that the compound nucleus, which is in air excited state (excitation 

energy above the ground state = incident energy T binding 

energy of oire particle), will live until this excitation energy, or a 

reasonable fraction of it, is concentrated again on one particle. 

3. The final break-up of the nucleus is independent of the mode 
of formation, i.e., regardless of how- the nucleus was formed there 
will be definite probabilities for decay into each of several possible 
residual nuclei. This can be explained in terms of the long life- 
time of the compound nucleus during which, complete statistical 
equilibrium, is assumed to be established — thus the nucleus forgets 
how it was formed; formation and disintegration can be regarded 
as independent events. 

For example, the ordinary Al nucleus ( l3 Al 2 ') can be formed as 
a "compound nucleus" in a highly excited state from any of the 



(199) 



reactions ; 


u Na 23 + W -» 1S A1 37 excited" 




12 Mg 2E + Tl 2 -> 13 A1 27 excited 




12 Xlg 2fi + H 1 -» "AF excited 




"Al 27 + t -> IS AF excited. 



The compound nucleus can then decay back, reversing the reac- 
tion, into any of the nuclei just mentioned, or also into AF + %, 



THE COMPOUND NUCLEUS 



1.11 



with a definite probability for each which is the same for all modes 
of formation. The residual nuclei may also be left in excited 
states, with probabilities which are also independent of the manner 
of formation. 

Formation of Compound Nucleus. The cross section for forma- 
tion of the compound nucleus cy may be written in the form 

H = tvR.% (200) 

where H, is the nuclear radius, and £ is a useful parameter, called 
the sticking probability, which is defined by this equation. 

For fast nuclear particles, i.e., X «R{\ ~ 10" 12 cm for 200-kv 
neutrons), the classical geometrical approach is valid since the 
uncertainty in position of the particle is only X. The cross section 
for capture of fast nuclear particles is certainly not greater than 
vB 2 since the interaction is negligible ii the particle passes at a 
distance from the nucleus. For slow neutrons, however, cross 
sections greater than irR 2 are possible since the position of the 
particle is poorly defined. To get a sticking probability which is 
always <1, the definition is revised. We define the contribution 
ffL to" the cross section due to particles of orbital momentum I, 

andset *, - (21 + 1)*X% (201) 

Then from general principles of quantum mechanics, £j must be 
less than (or equal to) 1. Moreover, equation 201 reduces to 
equation 200 for high energy since all values of I up to R/X will 
contribute appreciably (cf ." Chapter IX, p. 38) ; £ is a weighted 
average of % Neutrons were used in the above discussion to 
avoid questions involving penetration of the potential barrier 
winch would arise for protons and a- particles. 

The Bohr statement, that, any particle which hits the nucleus 
is eatight, is given more precisely by the equation 

In other words, the sticking probability approaches 1 at, high 
energies. This statement has been checked experimen tally with 
high -energy neutrons especially by Amaldi and co- workers, by 
Sherr, and by Graham and Seaborg. They find cross sections of 
about it/ & irft 2 , with E given, by a formula similar to equation 3, 
in good agreement with other methods of determining nuclear 
radii (sec Chapter II). 



112 



TOPICS NOT RELATED TO NUCLEAR FORCES 



Disintegration of Compound Nucleus. The probability that the 
compound nucleus will disintegrate in a particular way is related 
to the cross section for the corresponding inverse capture process 
with some factors containing Ihe density of initial and final states. 
This follows from considering a statistical equilibrium condition 
between the compound nucleus and all the possible states of all 
the residual nuclei into which it can disintegrate (similar to 
Chapter XI, p. 60). In. equilibrium, the number of nuclei present 
in a small energy range between E and E + dli will be proportional 
to the density of states p(E) in that energy range, and to a Boltz- 
mann factor. Since energy is conserved in the total system, the 
Boltzniann factors cancel out and the condition for equilibrium 
takes the form 

PjtWx-fji « m&B^jk (203) 

where p A and p Ti are the densities of initial and final states of the 
system at corresponding energies, and the W's represent prob- 
abilities for the direct and inverse processes. 

For our process, A is the excited compound nucleus 'rath a 
density of states pa(Ea) = I/Aaj where D is the average separa- 
tion between neighboring states, at an energy E A above the ground 
state of A- (Each state is counted according to its statistical 
weight.) Wa-*b is &hs probability of disintegration of the com- 
pound nucleus into a definite state of the residual nucleus B with 
energy E$ above its ground state, with the emission of a particle 
(say neutron') of energy E. We^a is the probability that nucleus 
B will capture this particle of energy E and produce a compound 
state of excitation E A . Finally, ph(Ejj) gives the number of states 
between E and E + dE available for tin: outgoing particle, viz. 

47T-/J 2 



PS 



v(2*Ky s 



(203a) 



with p and v the momentum and velocity of the outgoing particle. 
We now use the relation between the capture probability and the 
capture cross section, which is 



Wg^j 



vc f (E) 



(204) 



for one neutron in a. box of unit volume moving with velocity 
v => (2Efm) ' /2 , and the relation between the excitation energies E A 



and Eg, 



Ea 









E - B 






(205) 



THE COMPOUND NUCLEUS 



113 



where E is the energy of the outgoing particle and B its binding 
energy in the unexcited nucleus A. 

Using all the relations just given, and setting I = in equation 
201 (other I give veiy similar results), we now have a relation by 
nieans of which the disintegration probability Wa~*8 — iVh can 
be computed in terms of the sticking probability £g for the inverse 
capture reaction : 

n//;bi)(TVh) = m *$fe ( 20e ) 

or, inserting 203« and simplifying: 

T b /Da = fr/2* ( 206a ) 

This important equation relates the disintegration probability 
Fb, leading to a definite state of the residual nucleus, to the level 
spacing D A . For high energies, 6s approaches 1; for low energies 
it is proportional to the velocity v of the emitted particle. Both 
Da and T B can be deduced from experiment; D A and % B can also 
be estimated from various statistical models for heavy nuclei 
(Nuclear Physics B; Weisskopf, Phys. llcv. 52, 295, 1937; 57, 
■172, 1940). 

The disintegration probabilities tjj/fi >"'e also related to the 
widths of the resonances observed in these reactions: since; the 
total decay probability is 



V/h 



(i/h)^ 



B 



the time dependence of the wave function is of the form 



-iEl/h -lt/2h _ (J -i{li!- 



f]'lf/h 



(207) 



(208) 



(Note that the absolute square of the wave function gives the 
occupation of the state and decays according to equation 207.) 
Equation 208 has a Fourier transform * the absolute square of 
which is: 

(209) 






(E' ~E) 2 + (r/2) £ 



Thus T has the same dimensions as E and gives the width at half- 
maximum of the level, or resonance line. The quantity T B repre- 

* Taking the Fourier t ns.n?f mid with respect to time of a- time-dependent 
wave function gives the wave function 4i{B') in energy space. 












__ 



114 



TOPICS NOT RELATED TO NUCLEAR FORCES 



sents a partial level width, i.e.. the contribution to V arising from 
the disintegration into a definite end state B. 

Since the compound nucleus must eventually decay, the cross 
section for a reaction ending in state B is given by the cross section 
for forming the compound nucleus, times Fg/F. Thus 

(210) 



and for fast particles : 



ays = °7 iwr 



ujb = rS'| Tg/T 



(211) 



CONCLUSIONS ABOUT NUCLEAR REACTIONS 



Energy Distribution of Emitted Particles. From equation 206a 
we see that V B is almost the same for any final state B, since the 
sticking probability k B is a slowly varying function of the energy 
of the outgoing particle. This information is useful in predicting 
the energy distribution of the emitted particles. For example, 
if we consider the inelastic scattering of neutrons 



Z A + n->Z A -'^Z A ^n 



(212) 



and make use of the fact that the density of states in the residual 
nucleus increases rapidly with excitation energy, then we see that 
the residual nucleus will most likely be left in a fairly high excited 
state and the emitted neutron, will come out with low energies. 

The fact that emitted neutrons come out with greatly reduced 
energies has been experimentally confirmed for many target nuclei. 
Lead forms a notable exception to this rule. The reason for this 
may be that the first excited state in this instance is quite high — 
so that this rule would not be confirmed unless higher energy 
incident neutrons are used. In fact, the incident energy must 
be high enough so that the residual nucleus B possesses a great 
many levels with an excitation energy less than the incident 
kinetic energy E , in order that the statist! eal considerations used 
may be valid. 

Shadow Scattering. In neutron-scattering experiments a purely 
wave-optical effect must be considered at high incident energies 
(X « B). for which we have said the capture cross section is x.8 . 
In this case, the nucleus can be regarded as a black sphere of 
radius R which casts a shadow. This is described in the language 
of wave optics by saying that just enough light is scattered in the 



THE COMPOUND NUCLEUS 



1 15 



forward direction to cancel the incident beam. This would mean 
a cross section for shadow scattering of irR 2 . Fur therm ore, to 
cancel the incident beam behind the sphere, this shadow scattering 
must be of the same energy, i.e., it represents elastic scattering. 
According to an elementary wave-optical argument, the shadow 
Scattering will be mostly confined to an angle X/fi from the 
forward direction. 

In the case of light, for which normally X « B, the shadow 
scattering is not easily measurable since the shadow extends prac- 
tically to infinity. In the nuclear case %/B is, say, y 6 or %, so 
that the umbra or region of complete shadow extends only a short 
distance back of the nucleus, certainly not as far back as the 
measuring apparatus. Thus il is possible to make measurements 
outside the main beam but still at small enough angles to it to 
obtain the elastic shadow scattering. The existence and general 
features of shadow scattering have been confirmed experimentally 
by Kikuehi ot ah, Amaldi et ah, and Backer. 

Charged Particles. The emission of charged particles such as 
protons requires the penetration of a potential barrier. This 
penetration probability is similar to that given in the theory of 
a-decny and is quite small unless the emitted protons have energy 
nearly equal to, or greater than, the barrier height B. Thus, in a 
rough way, we may say that the protons must leave with a mini- 
mum energy B. This would leave the residual nucleus at a lower 
energy than if neutrons were emitted. Since the density of residual 
nucleus states decreases rapidly with decreasing energy, the: 
probability for proton emission will be much smaller than that for 
neutron emission because of the fewer number of states available, 
especially if the nuclear charge is high and the available energy low. 

y-ray$. The emission of 7-rays will in general be small com- 
pared to heavy particle emission when the latter is energetically 
possible because the coupling of the nucleus with the radiation 
held involves the small factor e 2 /hc = 1/137, 



DENSITY OF NUCLEAR ENERGY LEVELS- 
NUCLEAR TEMPERATURE 



The density of nuclear energy levels increases rapidly as a 
function of energy. To see how this comes about a model which 
is only a crude: approximation is used. We consider the nuclear 



110 



TOPICS NOT RELATED TO NUCLEAR FORCES 



particles as independent of each other, and suppose each of them 
has a set of equally spaced energy levels spaced by an energy 
difference A. Then, the excited states of the .system will also be 
spaced by the Interval A, and will have a greater statistical weight 
the greater the excitation energy, because of the greater number 
of ways of dividing the energy among the particles. When an 
interaction among the particles is then introduced, there will be 
splitting of each energy level ; and the statistical weight of an energy 
level of the non-interacting system is a measure of the energy level 
density in the same region of the spectrum, after the interaction 
has been introduced. 

To calculate the level density a model of the nucleus must be 
used. Four models will be mentioned. (For more details set: 
Nuclear Physics B, p. 79.) 

1. Free Particles in a Box of the Size of the Nucleus. The level 
spacing D is proportional to exp( — V E), where E is the excitation 
energy of the nucleus. For A = 120, E = 3 Mev, we get D ~ 10 
ev, which is about what is observed. 

2. Free Particle in a Box, with Correlations. Bardeen has 
pointed out that the free particle model must be modified to be 
in accord with the assumption of exchange forces. The result 
gives a level spaaing depending on excitation energy in about the 
same way as before, but the level spacings are somewhat wdder: 
D ~ 100 ev for A = 120, E = 8 Mev. 

3. Lattice Model. This model is the opposite extreme of models 
1 and 2, for the particles are here supposed to be firmly bound 
and capable only of small vibrations about equilibrium. The re- 
sults are similar to those for models 1 and 2. The level spacing is 
proportional to expf-J^). For A = 120 and E = 8 Mev, 
t>~ 100 ev. 

4. Liquid Drop Model For heavy nuclei this model is quite a 
good approximation. The level spacing is proportional to 
exp(-i? V; ) for small S and exp(-/i H ) for larger E. For A = 120 
and E = 8 Mev, D — 10 ev. 

All these models give a level spacing which is a decreasing func- 
tion of the energy of the form expL— /(£)], where f(E) is a slowly 
variable function of the energy. 

If the density of states, p{E) = 1/D, of any system is given 
as a function of energy then an entropy can be defined as 



THE COMPOUND NUCLEUS 



117 



£ = k log p(E), and a temperature as dS/dE = \/T(E). Each of 
the four models mentioned will therefore define a nuclear tempera- 
ture lis a function of excitation energy. It turns out that for 
10 Mev excitation energy, hT is of the order of 1 Mev, i.e., 
T = I0 10o K. 

The most satisfactory treatment of nuclear thermodynamics 
(Weisskopf, Phys. Rev. 52, 295, 1937) avoids a model and supposes 



I) = C exp(-L'Vi') 



(213) 



The constants B and C are determined from experiment: For low 
excitation energies the exponential is close to 1 so that D is about 
equal to C. From the observed position of the lowest excited levels, 
it is found that: 



For light nuclei (A ■ 
For heavy nuclei (A 



20) 
-200) 



a- 



io H 

10 3 



(214) 



B can then be determined from neutron resonance levels near 

E rv 8 Mev (binding energy of neutron in nucleus); this gives 

about : 

B = 2 for light nuclei 

(215) 
B = 4 for heavy nuclei 

if E is measured in Mev. 

Any of the level density functions lead approximately to a 
Boltzmann distribution for inelastically scattered neutrons. If 
the incident energy of the neutrons is Bq and the energy of the 
emitted neutrons is IF then tin.: excitation energy of the residual 

nucleus is E n - W. Supposing that, the level density of the 
residual nucleus is exp [+/(£)] and expanding, 

m =/(e^ -rmw+ ■■■ (m 

w : e get a level density 

exp f(E) - exp f(E q ) X exp( -fW) (21 7) 

Therefore, setting /' = 1/lcT (which is exactly the expression 
demanded by dS/dE = l/T) gives a Boltzmann distribution for 
the level density of the residual nucleus as a function of W and 
therefore for the kinetic energies of the emitted neutrons. A more 
careful consideration gives a probability of emission proportional 



US TOPICS HOT RELATED TO NUCLEAR FORCES 

to VW «K P (-W/kT) or W ^ 9 ^-W/kT) but experiment lias 
not as yet given enough data to make it possible to distinguish 
between them. 

RESONANCE PHENOMENA 
Let the energv levels of a nucleus Z A be as shown in Fig. 10 and 
consider the process 7^ + i> - Z A U the incident neutron 
lias exactly the right energy to form Z A in one or its excited states 
the probability of capture is large. Such energies are called 
resonance energies of the compound nucleus. The experimental 



Vp- Resonances 



_Binc!ing energy 
""'of neutron 



-^^.Ground state 




Fig. 16. Energy levels 
of a au&leus, 



Pig. 17. Typical experimental 

cross -seetiuii of a nucleus for 

slow neutrons- 



evidence (see Fig. 17) for neutron resonance energies m capture 
processes led to the first theories of the compound nucleus .ex- 
perimental ly, for A ~ 100, the level spacing D is about 10 ev, if 
E is about the binding energy of the neutron, i.e., 8 Mev. D is 
about the same at A ~ 200, and the appropriate binding energy 
E ~ 5 Mev Tins can be understood because, on the one hand, 
the number of particles is greater (and thus there are more possi- 
bilities of distributing the energy) ; on the other hand the excita- 
tion energy (binding energy of the particle) is smaller. For A 
smaller than 100, the level spacing increases rapidly. 
' There are several nuclei for which m0 re than one resonance is 
knowm Among elements having only one (abundant) isotope, 
In has 3 resolved resonances, I lias 5, and Ta 7. In addition, 
many other elements show more resonances than isotopes. .Most 
of the experimental evidence was obtained by Rainwater, Havens, 
and their collaborators, in several papers in Phys. Rev. 71 £1947). 
In some; cases, onlv one resonance is observed; the level spacing 
is then not directly known but it pan be taken as of the same order 
of magnitude as the kinetic energy of the neutrons corresponding 
to the first resonance. 






THE COMPOUND NUCLEUS 



11!) 



For protons, capture resonances have been observed only for 
the very light nuclei. The level spacings are of the order of 10 
to 100 kev with an excitation energy of -10 Mev. lor heavier 
nuclei the Coulomb barrier prevents capture resonances for 
protons because the excitation energies which result after a proton 
lias been given sufficient energy to get over the Coulomb barrier 
are so high that the resonance levels overlap. A few resonances 
have also been observed for a-particles, the reactions of which 
lead mostly to the emission of protons or neutrons. 

The width T of a nuclear energy level is defined as Y = n, r, 
where r is the lifetime of the level. For most of the slow neutron 
capture levels the width is about 0.1 ev. This can be decomposed 

r=r 7 + r vl (218) 

into the neutron width and the y-ray width. Almost all of r is 
T which means that capture is far more probable than scattering 
for slow neutron resonances.* T, t may be determined separately 
in two different ways. First, the capture cross section at exact 
resonance is given by 



(const) X TjE r T 



(219) 






p is the width of the resonance at half-maximum; therefore, T\ 
can be determined from , at resonance, T and E T , Second the 
ratio of scattering to capture cross sections at resonance is iyi T , 
and f, is very nearly equal to i\ Unfortunately, in order to get 
the scattering cross section at resonance it must be ^entangled 
from the potential scattering (Nuclear Physics B, p. 162) so that 
this second method is ordinarily not ot much use. . 

The first experiments on neutron capture were done by fiermi 
and his collaborators, and by Moon and Tillman, using an ingen- 
ious but rather complicated method: a neutron beam from which 
the thermal neutrons had been removed by a cadmium absorber 
impinged on an indium detector. Comparison of the radmactivi- 
ties produced in this detector with and without an indium absorber 
intervening, showed that neutrons which activated the indium 
detector were strongly captured by the indium absorber It a 
silver absorber was used instead, the absorption was small. On 

* Mn has a Strong resonance at about 300 ev which gives mostly scattering 
^"therefore has r n » iy TWs is lo be expected for light nuele: because 
of their large level spacing; see equation 206a.-Ae(e added m prwf. 






120 



TOPICS NOT RELATED TO NUCLEAR FORCES 



the other hand, a silver detector showed about as much radio- 
activity with and without the indium absorber, but with a silver 
absorber the beam was very strongly attenuated. The conclusion 
was that indium and silver were activated by neutrons of two 
different energies. At present, the most satisfactory method con- 
sists in using a modulated cyclotron beam and determining the 
velocity of the neutrons by their time of flight to tin- target. For 
very slow neutrons, a pile and a crystal spectrometer are often 
preferable. 

THE DISPERSION FORMULA 

Breit and Wigner were the first to develop a theory of nuclear 
resonance processes. The result was analogous to that in the 
theory of optical dispersion 

n - ■ 1 -j, (220) 

The measurements using velocity selection can check the shape of 
this curve and at the same time determine E T and P. To get the 
coefficient of proportionality in equation 220, suppose that the 
cross section cr is for the production of B with A incident. Then, 
since the cross section is proportional to the half-width for disinte- 
gration into B, it must contain fs, It also must contain T A for 
symmetry reasons. This follows from the principle of detailed 
balance: apart from statistical weights and a factor depending on 
the ratios of momenta, o- A _> B should be equal to ff B -,A- (See 
Chapter XL) 

Finally we know that for the simplest case in which only one 
kind of particle can be emitted or absorbed, T A = Fs = F, and 
we know further that in this instance the largest possible cross 
section for particles with I = is IttX 2 . Clearly, in the general 
case, the wave length of the incident particle must occur. Collect- 
ing all information, 



a = %X, 2 



Va Tu 



(221) 



(S - E r f + (l'/2) 3 

This is known as the one-level Breit- Wigner formula. It gives 
the correct dependence on momentum, in accord with the prin- 
ciple of detailed balance 






THE COMPOUND NUCLEUS 



121. 



For the dependence on the spin of the compound nucleus and 
the generalization to more than one resonance level see Nuclear 
Physics B, p. 101. There is only one instance in which the many- 
level formula has been of use, namely, 



He 4 + n -* He s -» He 4 + ft 



(223) 



which has two partly overlapping resonances near 1 Mev. 

The dispersion formula has been derived many times. The 
derivation must be quite different from, the treatment in optics, 
where the interaction of the incident light and the atom can be 
taken as a small perturbation. 

For high-energy neutrons the dispersion theory goes over into 
the statistical theory given previously. The partial widths of 
the levels become of the order of magnitude of the level spacing 
and the resonances are no longer observable. 

For extremely slow neutrons, well below the first resonance, 
Ta. is proportional to v (this follows from the fact that F A is propor- 
tional to the density of states in momentum space, j/{dp/dE) rv -p) 
and so the Breit-Wigner formula reduces to 



X 2 v rv 1/s 



(224) 



This is the well-known .1 fv law for the cross section at very low 
energy. It makes the number of processes per second, which is 
av, independent of the energy distribution and proportional only 
to the total particle: density. For very light nuclei, the spacing D 
is very large and the 1/v law holds up considerable energies. For 
gin + ?l ^B" it is valid to 50,000 ev. Absorption by B 10 is 
therefore used for measuring neutron velocities. 



APPENDIX: TABLE OF NUCLEAR SPECIES 



Explanation" of the Table 



Column 1: 
Column 2: 
Column 3: 
Column 4: 

Column 5: 



K 

e~ 



Z." Atomic number of the element. 

Element." Chemical symbol of element. 

A." Mass number of the Isotope. 

Abund., per cent." Per cent abundance of isotope in the 

afurally occurring element. 

Disintegration." Symbols for nuclear processes are: 

1 isomeric transition. (Emission of ->-rays or conversion 
electrons.) 
Electron capture. 
Internal conversion electrons. 
Negative, positive beta-particle emission. 
a Alplia-particle emission. 
n, H Emission of neutrons, protons. 

V Denotes that the paitieular isotope has not been identi- 
fied with complete certainty. Parentheses enclosing 
one or more activities denote uncertainty in these, but 
not in the identification of the isotope to which they 
are assigned. Thus, 4; Ag 103 has been classified and found 
definitely to have fi" activity; however, it is not certain 
that 47 Ag 108 also has K-capture and conversion .elec- 
trons. A comma setting off le" from one or more 
symbols indicates that the conversion electrons belong 
to the isomeric transition. 
Columns and 7: Masses, with probable errors. A value in parentheses 
indicates that the mass has been estimated from theory, the 
isotope not having been produced as yet. 
Column 8: Spin of the designated isotope. 



Main Reffrenous 

G. T, Scaborg, Table of Isotopes, Rev. Modern Phys. 16, I, 1944. 

E. Segre, Isotope Chart, issued by Los Alamos Scientific Laboratory, 1 9*8, 

In general, isotopes classified as A to D by Seaborg and Segre have been 

included in this table, i.e., all those for which at least the assignment to a 

definite element is certain. 



123 



121 




Abund., 


APPEN 
Disinte- 


DIX 


Error 




Z Element, 


A 


per eeat 


gration 


Mass 


X UJ 6 


Spin 


n 


1 






1.008 93 


3 


H 


1 H 


1 


99.98 




1.008 123 


O.fi 


H 




2 


. 02 




2.014 70S 


1.1 


i 




3 




(T 


3.017 02 


3.4 


H 


2 He 


a 


~10~ 5 




3.017 00 


4 






4 


100 




4.003 90 


3 







5 




n 


5.013 7 


35 











r 


6.020 9 


50 




3 Li 


5 




ii 


(5.013 6) 


60 






6 


7.5 




0.016 97 


5 


l 




7 


92 . 5 




7.018 22 


6 


n 




8 




f 


8.025 02 


7 




4 Br 


() 






(6.021 9) 


100 






7 




K 


7.019 16 


7 






8 




tx 


8.007 85 


7 






9 


100 




9.015 03 


6 


% 




10 




p- 


10.010 77 


8 






11 






(11.027 7) 






5 B 


9 






9.016 20 


7 






10 


18.4 




lO.Olfi 18 


9 


i 




11 


81.6 




11 .012 84 


8 


H 




12 




r 


12.019 


70 






13 






(13.020 7} 






6 C 


10 




(? + 


10.021 


30 






11 




[i+ 


11.014 95 


9 






12 


98.9 




12.003 82 


4 







IS 


1.1 




13.007 51 


K) 


H 




14 




p- 


14.007 67 


5 






15 






(15.016 5) 






7 N" 


12 






(12.023 3) 








13 




+ 


13. 009 88 


7 






U 


99 . 62 




14.007 51 


4 


i 




15 


0.3S 




15,004 89 


21 


H 




Hi 




r 


> 16. 006 5 
< 16.011 








17 






(17.014) 






8 


14 






(14.013 1} 








15 




v + 


15.007 8 


40 






IS 


99.757 




16.000 000 


Standard 







17 


0.039 




17.004 50 


a 






IS 


0,204 




IS. 004 9 


40 






19 




P~ 


19.013 9 







TABLE OF NUCLEAR 3PECIE8 















Abund., 


Disinte- 




Error 


z 


Element 


A 


per cent 


gration 


Mass 


X 10* 


9 


F 


16 






(16.017 5) 








17 




ff* 


17.007 S 


30 






18 




.3+ 


18.006 5 


00 






19 


100 




19.004 50 


26 






20 




r 


> 20. 004 2 
<20.009 2 








21 






(21.005 9) 




10 


No 


18 






(18.011 4) 








19 




+ 


19.007 81. 


20 






20 


90.00 




19.998 77 


10 






21 


0.27 




20.999 03 


22 






22 


9.73 




21.998 44 


30 






23 




r 


23.001 3 




11 


Na 


21 




p + 


21.003 5 








22 




$+ 


21.999 9 


50 






23 


100 




22.996 18 


31 






24 




a~ 


23.997 5 


45 






25 




Ufr 


(24.996 7) 




12 


Mg 


22 






(22.006 2) 








23 




ft + 


23.000 2 


40 






24 


77.4 




23.992 5 


00 






25 


11.5 




24.993 8 


90 






26 


11.1 




25.9S9 8 


50 






27 




r 


26.992 8 


150 


13 


Al 


25 




3 + 


24.998 1 


100 






20 




P + 


25.992 9 


150 






27 


100 




26.989 9 


80 






28 




&~ 


27.990 3 


70 






29 




P~ 


28.989 3 


80 






30 






(29.995 4) 




1.4 


Si 


27 




P + 


26.994 9 


90 






28 


89.6 




27.986 8 


60 






29 


6.2 




28.986 6 


70 






30 


4.2 




29.983 2 


90 






31 




P~ 


30.986 2 


(iO 






32 






(31.964 9) 




15 


P 


29 




e+ 


28.991 9 


100 






30 




p + 


29.9S7 3 


10 






31 


100 




30.984 3 


50 






32 




r 


31.9S2 7 


40 






33 






(32.982 6) 








34 




$- 


33.9S2 6 


40 



125 



Spir 



H 



; 



126 








APPENDIX 
















A bund., 


DltfMLilV 






Error 




Z 


Element 


A 


per cent 


gration 


Mass 


X 10 s 


Spin 


m 


S 


31 




p + 


30.989 9 










32 


95.1 




31. 


980 89 


7 









33 


0.74 




32. 


980 


GO 








34 


4.2 




33. 


977 10 


33 








35 




F 


31. 


978 8 


SO 








38 


0.016 




35. 


97S 


100 








37 




§~ 


36. 


982 1 


30 




17 


CI 


33 




# 


32 


986 










34 




P 


33 


980 1 


200 








35 


75.4 




34 


978 67 


21 


H 






36 




<3 + ,S" K 


35 


97S 8 


100 








37 


24.0 




36 


977 50 


11 


% 






38 




fr 


37 


981 


300 








39 






(3S 


979 4.) 






18 


A 


35 




r 


3-1 


985 










36 


0.307 




35 


978 


100 








37 




K 


36 


977 7 










38 


0.061 




37 


974 


250 








3D 






(38 


975 5} 










4!) 


99.632 




39 


975 6 


60 








41 




r 


10 


977 


60 




19 


K 


37 






(36.983 0) 










38 




3 + 


37 


.979 5 










39 


93.38 




38 


.974 7 




% 






40 


0.01.2 


e-x 


39 


.976 


100 


l 






41 


6.61 




10 


.974 




% 






42 




w 














■13 




Ufr 










20 


Ca 


39 




VP* 














40 


96.96 




39 


.975 3 


150 


(.) 






41 




TJ K g- 














42 


0.64 




41 


.971 1 










43 


0.15 




■12 


.972 3 










■11 


2.06 
















■15 




r 














46 


0.0033 
















48 


0.19 
















49 




r 










21 


Bo 


41 
43 
It 


















45 


100 




44 


.966 9 


so 


% 






46 




S~ if 











Z Element 

21 Sc 
{cant.) 

22 Ti 



23 V 



24 Cr 



25 Mn 



26 Ee 



27 Co 



TABLE OF NUCLEAR SPECIES 



A 
47 
48 
49 

15 
46 
47 
48 
49 
50 
51 

47 
48 
49 
50 
51 
52 

49 
50 

51 
52 
S3 
54 

55 

51 

52 
54 
55 
56 

53 

54 
55 
56 
57 
58 
59 

55 
56 
57 
58 
59 
60 



Abund,, 
per cent 



7.95 

7.75 

73.45 

5.51 

5.34 



100 



4.49 

83.78 
9.43 

2.30 



100 



0.04 

91.57 
2.11 
0.28 



Disinte- 
gration 

r 

(3+ 



U ,3 '" 
£+ K 

a k 

r 

U K cT 



V 

f3-'~K 
K 



100 



a+ k 
$+ K r 
13+ K 



Mass 



45.966 1 
46.964 7 

47.963 1 

48.964 6 
49.952 1 
50.958 7 



50.957 7 



50.958 
51.956 

52.956 



54.957 



53.957 



Error 
X 10 5 



127 



Spin 



100 

100 

50 

60 

40 

100 



50 



% 



55.956 8 170 

56 . 957 



n 



128 






ArPENDIX 










Abund., 


Disinte- 




Error 


'/, EI c men t, 


A 


per cent 


gration 


Mass 


X 10 & 


28 M 


57 




& 








58 


67.1 




57.959 4 


10 




59 




V .a ' 








GO 


26.7 




59.949 S 


40 




61 


1.2 




60,953 7 


150 




62 


3,g 




61.949 3 


40 




ea 




r 








0-1 


0.88 




63.947 1. 


60 


29 Cu 


58 
BO 
01 
62 




e+ 

0+ K 

P + 








63 


70.1,3 




62.957 


400 




G4 




r& K 








56 


29.87 




64.955 


400 




66 




r 






30 Zn 


63 




^ 








64 


50 . '.) 




63.955 


400 




65 




fl • K f 








m 


27.3 




65.954 


400 




67 


3.9 




66.954 


400 




68 


17.4 




67.955 


300 




69 




li~ I 








70 


. 5 




69.954 


300 


31 Ga 


(it 
65 

m 

67 
68 




U 3+ 
K '~e~ 
fi + 
K ,.r 








69 


61.2 




68 . 952 


800 




70 




8~ 








71 


38.8 




70.952 


900 




72 




0~ 








71 




V fi- 






12 Ge 


69 

70 
71 
72 
73 

74 
75 
76 
77 
78 


21.2 

27.3 

7.9 

37.1 

fi.5 


ll 

3+ A* ( - 

«r 







Spin 



H 



% 



% 



TABLE OF NUCLEAR SPECIES 



Mass 









AbuTiJ., 


Disinte- 


z 


Element 


A 


per cent 


gration 


33 


As 


72 

73 
74 
75 
76 
77 
78 


100 


U 3+ 

V P + K e 

r ? + 

&- /3 + K 
Up~ 


34 


He 


74 

75 
76 
77 
78 
79 
80 
82 
83 


0.9 

9.5 

8.3 

2-1.0 

48.0 
9.3 


K e" 


35 


Br 


78 

79 
80 
SI 
82 
83 
84 
85 
87 


50-6 
49.4 


3~, 1 <T 

r 

r 
r 



129 

Error 

X 10 5 Spin 



% 



M 



m 


Kr 78 
79 
80 

81 
82 


0.35 
2 01 

1 I . 53 


(./ 1 <~ 






83 


] 1 . 53 


i «r 


% 




84 


57.11 








85 




p~ 






86 


17.47 








87 




u tr 






88 




0- 






89 




,a- 






90 




U fi- 


* 




91 




ll fr 






92 




ur 






94 




U fi- 






95 




ll 07 




^EJ ' 








J^™ 



130 






Abund., 


APPENDIX 

Disinte- 


Z 


Element 


A 


per cent 


gration 


37 


Rb 


82 
84 
85 
86 


72.8 


V 
U 

0~ 






87 


27.2 


P~ 






88 




tr 






gg 




r 






'JO 




V fi- 






91 




ll rr 






92 




u&- 






!)4 




U ii~ 






95 




v$r 


38 


gt 


84 
85 
8(5 


0.50 
9.86 


J <r K 






87 


7.02 


le- 






88 


82.50 








89 




ft" 






90 




UfT 






01 




U fi- 






92 




ll fi- 






01 




ll ft- 






95 




ur 


39 


Y 


86 

87 
88 
89 
90 
91 
92 
04 
95 


100 


UK 
(l e~) K 

V fi~, t e- 
U fi- 
ll fi- 
ll r 


40 


Zr 


89 
90 
91 
92 

03 
94 
95 


48.0 
11.5 

22.0 

17.0 


fi~, I or B 
U0- 

v a~ 



■11 



Cb 



96 

07 

90 

91 
92 



Mass 



Error 
X 10* 



Spin 
% 



TABLE OF NUCLEAR SPECIES 



ur 

U8+ 

UK.tr 

W 







A bund., 


Disinte- 




Error 


Z Element 


A 


per tent 


gration 


Mass 


X 10 5 


41 Cb 


93 


100 








(sent.) 


94 

95 
96 

97 




a 

U 3~, I cr 

U 






42 Mo 


92 

03 


14.9 


U (3+ 








94 


9,4 




93.945 


800 




95 


16.1 




94.946 


800 




00 


J6.G 




95.944 


800 




97 


9 . 05 




96 . 945 


900 




08 


24.1 




97.943 


000 




09 




v r 








100 


9.25 










101 




u r 








102 




U fi- 






43 Te 


96 

98 

99 

101 

102 




ll K 

U K <T 
6", I e~ 
U fi- 
ll r 






41 Ru 


90 


5.08 




95.945 


1100 




08 


2.22 




97.943 


1100 




00 


12.81 




98 . 944 


1100 




100 


12.70 




99.942 


1100 




101 


16.98 




100.946 


1100 




102 


31.34 




101.941 


1100 




103 




r 








104 


18.27 










105 




fT 








106 




u r 








107 




O $r 






45 Rh 


102 




s-ft + 








103 


100 


I 


102.941 


1100 




104 




li~, lf~ 








105 




$r 








106 




U fir 








107 




rye- 






46 Pd 


102 


0.8 




101.941 


1100 




104 


0.3 




103.941 


1100 




105 


22.6 




104.942 


900 




106 


27.2 




105.941 


1000 




108 


26.8 




107.941 


1000 



131 



Spin 

% 



H 



132 








APPEXDIX 














Abund 


., Ksiiite- 




Error 




Z 


Element A 


per ecu 


t grat 


ion 


Mnss 


X 10 s 


Spin 


46 


Pd 


109 




u tr 










(cont.) 




110 
111 
112 


13,5 


pr 




109.941 


1000 




■17 


Ag 


105 
106 




U K 

3+ K B~ 














107 


SI. 'J 


{/O 




106.945 


600 


] A 






108 




r (K E 


") 












109 


48.1 






108.944 


700 


k 






no 




8~ 














in 




P~ 














112 




w~ 










m 


Cd 


106 
107 

108 
110 
111 
112 
113 
114 
IIS 
116 
117 
1.18 


1.4 

1.0 

12. S 
13.0 
24.2 
12.3 

2S.0 

7.3 


UK 
U, I ,- 










49 


In 


no 
.1 11 

112 
113 

114 
1.15 

no 

11.7 


4.5 
95.5 


U 8+ 

V a+ «- 

U, I e~, 
I er 

1 e~, 3~ 

1 <r 
a~ 


Ke~ 








50 


Sn 


112 
11.3 

114 


1.1 

O.S 


KtT 














115 


0.4 






114.940 


1400 


H 






116 


15.5 






115.939 


1400 








11.7 


9.1 






ne. 937 


1400 


H 






US 


22.5 






117.937 


1400 








110 


9.8 






118.938 


1400 


H 






120 


28.5 






119.937 


1400 








121 




U-ftr 














122 


5.5 






121.945 


1400 








123 




U8- 











TABLE OF NUCLEAR SPECIES 









Abund., 


Disinl.c:- 


Error 


z 


Element 


A 


per 


cent 


gration 


Mass X 10 5 


50 


Si) 


124 


6 


8 




123.944 1400 


{cmd.) 




125 
127 
128 






if fir 

U 3~ 
11 3- 




51 


m 


120 
121 
122 
123 
124 
126 
127 
128 
129 
132 
133 
136 


56 

44 




ti- 
ll fir 

&~ 
&~ 

U3~ 

u 3- 




52 


Te 


120 
121 


Q 


088 


IK<T) 








122 


2 


43 


(ier) 








123 





85 










124 


4 


59 










125 


6 


93 










126 


18 


71 










127 






0~, 1 e~ 








128 


31 


86 










129 






8~, I <r 








130 


34 


52 










131 






8~, I e~ 








132 






tiff" 








133 






u 3- 








135 






fi~ 








tae 






u tr 








137 




, 


u a~ 




53 


1 


124 
1 26 
127 
128 
130 
131 
132 
133 
135 
136 
137 


100 




3+ 

a~ K 

a- 
a- 
3- 
Vb- 

V 8- 

u 3- 





133 



Spin 






% 



J 



4 






APP1 

Abund., 


2NDIX 

Disinte- 


Error 


z 


Element 


A 


per 


cent 


gration Mass X 10 s Bpin 


54 


Xe 


124 
126 
127 

128 





1 


094 
088 

90 


U, I e~ 








129 


26 


23 




u 






130 


4.07 










131 


21 


17 




H 






132 


26 


96 










133 






&)lr 








134 


10 


54 










135 






(Tl 








136 


8 


95 










137 






U P~ 








138 






Up~ 








139 






sr 








140 






r 








141 






0- 








143 






u tr 








144 






Up~ 




55 


Qa 


130 
132 

133 
134 
136 
137 
138 
139 
140 
141 
142 
143 


100 




V 

v k <r 

Uff- 

u p- 

pr 

Up- 

P~ 

v tr 

u 


% 


50 


Ba 


130 
132 
133 
134 
135 





•2 
8 


101 
097 

42 
59 


I er 


% 






136 


7 


81 











137 


11 


32 




% 






138 


71 


.66 




G 






139 






p~ 








140 






p~ 








141 






p- 








142 






ur 








143 






ur 








145 






up- 





TABLE OF NUCLEAR SPECIES 



135 









Abund., 


Disinte- 




Error 




z 


Element 


A 


per cent 


gration 


Mass 


X 10 5 


Spin 


57 


L:i 


137 




U K 












139 


100 




138.953 


800 


K 






140 




r 












141 




P~ 












143 




U 3~ 












144 




up- 












145 




Up~ 








58 


Ce 


136 
138 
140 
141 
142 
143 
L44 
145 
147 


<1 

<1 
89 

11 


r 

Up~ 

u p- 
u p- 

U p~ 








59 


Pr 


140 
[41 
142 
143 
144 
145 
147 


100 


p + 

p~ 

u r 

up- 

Up~ 
UfC 






M 


60 


Nd 


141 

142 
143 
144 


25.95 

13.0 

22.6 


p + 












145 


9.2 




144.962 


400 








146 


16.5 




145.962 


400 








148 


6.8 




147.962 


400 








150 


5.95 




149.964 


400 




61 


61 


143 
144 

145 
146 

147 




Up~ 
U I or K 

Up~ 

u 
up- 








62 


Sm 


144 
146 
147 

1.48 
149 


3 

16.1 
14. 2 
15.5 


VI 

a 









136 






A 


PPEXDIX 








Abund., 


Disinte- 


2 


Elemenl 


A 


per cent 


gration 


62 


Sm 


150 


11.6 




(cont.) 


151 




u-fir 






152 


20.7 








154 


18.!.) 




S3 


Eh 


151 
152 
153 
154 
155 
156 
157 
158 


49.1 
50.9 


U fi~ e~ 

Utf 

V ti- 
ll ti- 
ll $- 


64 


Gd 


152 

154 
155 
156 
157 
158 
160 


0.2 
1.5 
18.4 
19.9 
18,9 
20.9 
20.2 




66 


Tb 


159 
160 


100 


fr 


OG 


Dy 


158 
160 
161 
162 
163 
164 
165 


XI. 1 

0.1 
21.1 
26.6 
24.8 
27.3 


it 


67 


Ho 


165 
166 


100 


u &- 


88 


Er 


162 

164 
166 
167 

168 
166 
170 


0.1 

1,5 

32.9 

24.4 

26.9 

14.2 


USi~ 


6ft 


Tm 


169 

170 


100 


03-) 



Mass 



Error 

X IB 6 



Spin 



m 



TABLE OF XUCLEAR SPECIES 



Mass 



153.971 


600 


154.971 


600 


155.972 


600 


156.973 


600 


157.073 


600 


159.971 


600 



% 



H 









Abund., 


Disinte- 


z 


Element 


A 


per cent 


gration 


70 


Yb 


168 
170 
17! 
172 
173 
174 
175 
176 


0.08 

4.21 
14.26 
21.49 
17.02 

29.58 

13.38 


U 


71 


Lu 


175 


97.5 








176 


2.5 


0- K 






177 




if ft 


72 


Hf 


174 
176 
177 
178 
179 
180 
181 


0.18 
5.30 
18.17 
27.10 
13.85 
35.11 


li~ 


73 


Til 


180 




(.3") IS. e 






181 


100 


I 






182 




ft 


74 


W 


180 
182 
183 

184 
185 
186 

187 


—0.2 
22 . 6 

17.3 
30.1 

29.8 


u p- 
w'fr 


75 


Re 


184 
185 
186 
1S7 

188 


38.2 
61.8 


UK 
U P~ 

u p- 


76 


Os 


184 
186 


0.018 

1.59 








187 


1.64 


K 






188 


13.3 








189 


16 . 1 








190 


26.4 








191 




v p- 






192 


41.0 








193 




Ujr 



Error 
X 10 s 



137 



Spin 



X 



>7 



<% 



<H 



7 A 



3k 






% 



189.04 2000 Viar% 

190.03 2000 

192.04 2000 



13S 






AI 
Abund., 


PENDIX 

Ilisinlr- 




Error 




Z 


Element 


A 


per cent 


graf.ion 


Mass 


X 10 s 


Spin 


77 


Ir 


191 
192 


3S.5 


r 


191 .04 


2000 


H 






193 


61. 5 




193.04 


2000 


% 






194 




P~ 








78 


Pt 


192 


0.8 














194 


30.2 




194.039 


1400 








195 


35.3 




195.039 


1400 


M 






198 


26.6 


(/O 


196.039 


1400 








197 




jjf 












198 


7.2 




198.05 


2000 








199 




r 








79 


An 


196 




TJ $T e~ 












197 


100 


I 


197.04 


1000 


% 






198 




§r e~ 












199 




l3~ 












200 




v&- 








SO 


Sg 


190 
197 
198 


0.15 

10.1 


Ke~ 













199 


17.0 


I er 






H 






200 


23.3 















201 


13.2 








U 






202 


29.6 















203 




ff'tr 












204 


0.7 















205 




ir 








81 


Tl 


198 

199 
202 




U K f 
V K "e~ 
U K t~ 












203 


29.1 




203-05 


2000 


H 






204 




Upr 












205 


70.9 




205 . 05 


2000 


% 






206 




ur 










AcC" 


207 




r 










ThC 


208 




r 










Tl 


209 




p- 










RaC 


210 




e- 








82 


Pfa 


203 




u $+ 












204 


1.5 




204.05 


2000 








205 




v, i r 












206 


23.6 




206.05 


2000 









207 


22.6 




207.05 


2000 


}£ 



TABLE OF NUCLEAR SPECIES 










Abi 


ind., 11 i si ii fce- 




Error 


z 


Element 


A per 


eent gration 


Mass 


X to? 


82 


Pb 


208 52.3 


208.05 


2000 


icont.) 




209 


8~ 








RaJD 


210 


tt 








AcB 


21] 


0- 








ThB 


212 


g" 








Pb 


213 


fr 








SaB 


214 


r 






S3 


Bi 


207 


K sr 










209 100 


209.05 


2000 




RaE 


210 


i'~ 








AcC 


211 


ft" a 








ThC 


212 


§T\M, 








Bi 


213 


p~ a 








Rati 


214 


f}" a 






84 


Po 


210 


a 








AcC 


211 


a 








ThC 


212 


a 








Po 


213 


a 








RaC 


214 


a 








AoA 


215 


<x 








ThA 


210 


(i~a 








Po 


217 


a 








RaA 


218 


r a 






85 


At 


21 I 


K a 






88 


An 


219 


& 








Tn 


220 


a 








Rn 


221 


a 








Rn 


222 


a 






87 


87(AeK) 


22;; 


v pr 






88 


AcX 


223 


a 








ThX 


224 


a. 








Ra 


225 


a 








Ra 


226 


a 








MsTbi 


228 


v~ 






m 


Ac 


227 


fi- <z 








MsTh 2 


228 


l3~ a 






90 


RdAo 


227 


a 








RdTh 


228 


a 








Th 


229 


a 








To 


230 


a 








UY 


231 


$r 







139 



:pm 



140 




APPENDIX 












Ab unci., 


Disinte- 




Error 




Z Element 


A 


per cent 


gration 


Mass 


X io 5 


Spin 


90 Th 


232 


100 


a 


232.11 


3000 




(cord.) Th 


233 




r 








UXi 


234 




&" 








91 Pa 


231 
232 
233 




a 

{j- - 






X 


m 


234 




r 








VX-i 


234 




wi 








92 U 


233 




a 








UII 


234 


0.00518 


a 








AcU 


235 


0.719 


a 








u 


237 




r 








UI 


238 


99.274 


a 


238.12 


3000 




u 


23'.) 




r 








93 Np 


234 
235 
256 
237 
238 
239 




K 
K 

r 

ix 
ST 

r 








94 Pii 


238 
239 




a 

a 








95 Am 


241 




a 








96 Cm 


240 

242 




® 









^ 



INDEX 



Allen, 22 

Allowed transitions ( h <?-decay), 97 
in intermediately heavy nuclei, 104 
in mirror nuclei, 103 
in naturally radioactive nuclei, 104 
lifetimes in, 102 
a-parlicle, rote of, 84 
m-parlicle emission, 14 
a-r&d inactivity, lifetimes for, 6 
Alternation, intensity, in band spec- 
tra, 18 
Alvarez, 24, 53, 54 
Amaldi, 91, 93, 115 
Angular distribution, of neutron- 
proton scattering, 39 
at high energy, 03 
of proton-proton scattering, 69 
Angular momentum, 38 
Annihilation, of electrons and posi- 
trons, 2 
of heavy particles, 2 
Arnold, 25, 28 
Atomic weight, 3 
Atoms, theory of, 23 
Axial vector, 107 

Backer, 115 
Backus, 102 
Bailey, 44 
Band spectra, 15 
Barrier, potential, 7 

penetration of, 115 
Bartlett force, 82 
Beams, molecular, 15 
Bennett, 44 
Benjstralh, 44 
Be 8 , 5 
^disintegration, 10, 97 

and neutrino, 20 

Fermi theory of, 97 



^-disintegration, Konopiriski-U hi en- 
beck theory of, 100 

Kurie plot of, 100 

of neutron, 105 

selection rules in, 99 
^-lifetimes, 102 
/3-spectrum, 20 
Bcihc, 105 
Binding energy, of deuteron, 29 

of neutron in heavy nuclei, 1 1 7 
Binding of proton in molecule, effect 

of, 47 
Block, 15,24 
Bohm., 44 
Born, 37 

Bom's approximation, 18, 109 
Bose statistics, 16 
Breit, 69, 70, 71, 120 
Breit-Wigrier formula, 1 20 
Bretscher, 60 
Brickwe.dde, 53 
Britlouirt, 7 
Erode, 94 
Bulk matter, internal energies of, SO 

Capture of neutrons, by protons, 00, 
79 
in heavy nuclei, 1 18 
Cenler-of-mass coordinates, 37 
Chadwich, 29, 60 
Champion, 92 

Charge, I 

Charged meson theory, 88, 95 

Charged particles, nuclear reactions 

involving, 8 
Chemical bond effect on neutron 

scattering, 47 
Chemical properties, 1 
Compound nucleus, 109 
disintegration of, 1 12 



141 



INDEX 



143 



142 



INDEX 



' 



fl I 

1 1! 
Ill 

, 
ill 



Compound n&ofcja, formation of, 111 

lifetime of, 110 
Compton wave length, 94 
Condon, 

Conservatism of energy, 5 
Constituents of nuclei, 8 
Cork, 100 
Coulomb field, 101 
Coulomb repulsion, 6, 8 
Coulomb scattering, 64, 65 
Critchjidd, 105 

Cross section, for nuclear reactions, 
8, 114 
for capture of neutrons by protons, 

B0, 79 
for photoelectric disintegration of 

deuteron, 56, 79 
for scattering, of neutrons by heavy 
nuclei, 7, 114 
of neutrons by protons, 37, 40, 79 
of protons by protons, 64 
geometrical, 7 

for capture by heavy nuclei, 111 
total, 40 

D-D neutrons, 42 
de Brogiie wave length, 8 
of electrons, 9 
of neutron or proton, 8 
do Brogiie wave number, 37 
Decay, £-, see ^-disintegration 
Decay processes, 13 
Dee, 40 

Density of nuclear energy levels, 115 
Depth of nuclear potential well, 32, 70 
Determination of force constants, 70 
Deuteron, excited slates of, 34 

ground state of, 29 

interaction of, with radiation, 56 

magnetic moment of, 26 

photo disintegration of, 79 

physical properties of, 25 

quadrupole moment of, 27 

singlet state of, 43 

states of, with tensor forces, 74 

virtual state of, 45 

wave function of, 33 
DeWirc, 53 



Dipole radiation, 13 
Dirac S-function, 95 
Disintegration, p, 10, 97 

nuclear, 13 

of compound nucleus, 112 
D i sin I .e gra li on probabi 1 i I i es , 1 1 3 
Dispersion formula, 120 
Dispersion theory, 121 
Distribution, angular, at high energy, 

63 
Dunning, 53 

Einstein's relation, 2 

Eisenbud, 69, 70, 87 

Electric dipole moments in nuclei, 

absence of, 75 
Electron energies, distribution of, in 

^-disintegration, 18 
Electrons, annihilation of, 2 
in nucleus, 5, 9 
spontaneous emission of, 1.0 
Electrostatic interaction of protons, 7 
Emission, of a-particles, 14 
of /3-rays, 10, 14 
of 7-rays, 13 
of heavy particles, 13 
of light quanta, 14 
of neutrons, 13 
of protons, 14 
Energy, 5 
conserved, 5 
kinetic, 5 
Energy distribution in inelastic scat- 
tering, 114 
Energy equivalent, 2 
Energy levels, nuclear, density of, 115 
Exchange and spin, relation between, 

84 
Exchange forces, 81 
and saturation, 83 
effects of, 82 

quantitative theory of, 87 
Excited states, of deuteron, 34 

of nucleus, 111 
Experiments, on neutron-proton scat- 
tering, 42, 91 
on photo disintegration, 60 
on scattering by para-hydrogen, 49 



Experiments, proton-proton, advan- 
tages of, 64 

East neutrons, 7 

Fast nuclear particles, uncertainty in 

position of, 111 
Feather, 60 
Fermi, 43 

Eermi interaction, 97 
Fermi statistics, 16 
Fisk, 81 

Forbidden transitions, half-lives, 106 
Force, Bartlett, 82 
Heisenberg, 82 
Majorana, 82 
Wigner, 82 
Force constants, determination of, 76 
Forces, exchange, 81 

quantitative theory of, 87 
non-central, 73 
nuclear, meson theory of, 94 
ordinary, B on-sat urati on of, 81 
prol.on-prol'.on, 64 
saturation of, 80 
short-range, 66 
Formula, dispersion, 1 20 
Free particle model, 116 
Frixch, 44 
Frostier, 94 
Fundamental particles in nucleus, G 

7-rays, emission of, 13 

Gamow, 6, 106 

Gamow-Teller selection rules, 106 

Geometrioal cross section, 7, 111 

Gurlach, 15 

Gilbert, 40 

Go'ppert^Mayer, 33 

Goldhaber, 29, 60 

Gordon, 94 

Graham, 60, 111 

Ground state, of deuteron, 29 

of He 2 , (54 
Gurney, 6 
Gyromagnetic ratio, 24 

Hafslad, 66 
Halban, 60 



Ilamermesh, 53 

Hansen, 15 

H ovens, 118 

Haxby, 20 

Heist'ttberg, 81 

Heisenberg force, 82 

H eider, 56 

He-\ 5 

Herh, 70 

Heydenburg, 66 

High-energy neutrons, experiments 

with, 91 
High-energy phenomena, information 

from, 62 
Hyperfine structure of spectra, 15 

Inelastic scattering, energy distribu- 
tion in, 114 
Information obtainable from high- 
energy experiment, 63 
Intensity alternation in band spectra, 

18 
Interaction, electrostatic, of protons, 7 

of deuteron with radiation, 56 
Interference between nuclear and 

Coulomb scattering, 64 
Invariance, against inversion, 73 

relalivistie, 107 
Inversion of coordinate system, 73 
Isobars, 2, 1 1 

of neighboring Z, 1.1 
stability of , 1 1 
"Isomer of nucleus, 13 
Isotopes, defined, 1 
Isotopic spin, 84 
Isotopic spin functions, 85 



Jentschke, 29 

K electron capture 
Kawne, 71 
Kellogg, 23 
Kemhle, 74 
Keener, 78 
Kerst, 70 
Kikuchi, 101 
Kimura, 29 
Kinetic energy, 5 



10, 14, 22, 108 



115 



mr 



144 



INDEX 



Klei a -Gordon equal, ion, 94 
Kmuypinski, 97, 103 

Konopinsld-Uhlenbeek theory, 100, 

104 
Kramers, 7 
Krugcr, 92 
Kurie plot, 100 

Ladenbnrg, 40 

Lande's formula, 28 

Lmmhlin, 92 

Lawson, 100 

Levels, n u clear energy, density of, 115 
width, experimental, 119 

Lifetimes, p-, 102 

in allowed transitions, 102 
in forbidden transitions, 105 

LigM quanta, emission of, 14, 119 

Liquid-drop model, 80 

Li 5 , 5 

Low-energy phenomena, information 

from, 62 
Lyman, 101 

Magnetic moments, 15 

of deuteroii, 25 
Majorana forte, 82 
iliti.j7i.fle/er, 29 
Mass, reduced, 37 
M.ass excess, 4 
Mass number, I 

Mass spectrograph, 3 
Mosses, 37, 66, 67 

Matter, bulk, internal energies of, 80 

Meson theory of nuclear forces, 88, 94 

Metastable state of nucleus, 13 

Molecular beams, 15 

Molecular velocity effect on neutron 
scattering, 49 

Moment, magnetic, IS 

quadrupole, of deuteron, 27 

Moon, 1 19 

Morse, 81 

Mott, 37, 66, 67 

Myers, 29 

Neutral meson theory, 88 
Neutrino, 20 



Neutrino, /3-disintcg ration and, 20 

experimental evidence for, 21 
Neutrino mass, 20 
Neutron, p-decay of, 105. 
physical properties of, 24 
slow, cross section of, 43 
wave length of, 7 
Neutron emission, 13 
Neutron-proton scattering, 78 
experimental results on, 42, 69 
at high energies, 91 
Neutron spin, 45 
evidence for, 45 
from scattering, 45 
Neutrons, capture of, by protons, 60, 
79 
Li-D, 42 
fast, 7 

in nucleus, 5, 8 

scattering of, by ortho-hydrogen, 
49 
by para-hydrogen, 49 
by protons, 37, 78 

at high energies, 91 
by protons bound in molecules, 
47 
thermal, 43 
Non-central forces, 73 
Non-saturation of ordinary forces, 

81 
Non-zero spin, nuclei of, 18 
Nuclear abundance, 12 
Nuclear charge, 1 
Nuclear constituents, 15 
Nuclear disintegration, 13 
Nuclear energy levels, density of, 1 15 

width, 119 
Nuclear force, between neutron and 
neutron, IS, 71 
between neutron and proton, 30 
between proton and proton, 64, 71 
meson theory of, 94 
saturation of, 80 
spin dependence of, 45 
Nuclear reactions, cross section for, 
114 
general theory of, 110, 114 
involving charged particles, 8, 115 



INDEX 



145 



Nuclear scattering, 7, 64, 67, 114 
interference between Coulomb scat- 
tering and, 64 
Nuclear species, table of, 123 
Nuclear spin, 15, 23 
Nuclear temperature, 115 
Nucleus, absence of electric dipole 
moments in, 75 
basic facts on, 1 
oompouTirl, L09 

disintegration of, H2 
formation of, 111 
lifetime of, 110 
constituents of, S 
excited states of, lit 
fundamental particles in, 5 
isomer of, 13 

magnetic moment of, 15, 25 
metas table state of , 13 
quadrupole moment of, 27 
residual, 111 
size of, 

stable, regularities in, 2 
Nuckolls, 44 
Number, mass, 1 

Octopole radiation, 13 
Grtho-hydrogen, scattering of neu- 
trons by, 49 

Packing fraction, 4 

Para-hydrogen, scattering of neu- 
trons by, 49 

Parity of wave function, 75 

Parkinson, 70 

Partial width of level, 1 1 4 

Particles, charged, nuclear reactions 
involving, 8 
heavy, annihilation of, 2 
emission of, 13 

Pauli principle with isotopic spin, 86 

Peierls, 78 

Penetration of potential barrier, 115 

Phase slrifts, 37 

for I ?s 0, 38 
Photodisintegration, 56, 79 

experiments on, 60 
Photoelectric effect, 56 



Photo magnetic effect, 57 
Physical properties, of deuteron, 25 
of neutron, 24 
of proton, 23 
Titter, 53, 54 
Plain, 70 

Polar four vector, 107 
Position of fast- nuclear particles, un- 
certainty in, 11 1 
Positrons, annihilation of, 2 

spontaneous emission of, 10 
Potential, relation between range and 

depth of, 32, 70 
Potential barrier, penetration of, 

115 
Potential well, rectangular, between 
two protons, 70 
of deuteron, 31 
Powell, 92 

Probability, disintegration, 113 
penetration, 115 
sticking, 111 
Proof of saturation, 84 
Properties, chemical, 1 
Proton, physical properties o) , 23 
Proton emission, 14 
Proton-proton e^Srka^tttS, advan- 
tages of, 64 
Proton-proton forces, 64 
Proton-proton scattering, 64 
experiments on, 70 
theory of, 65 
Proton resonances, 119 
Protons, bound in molecules, scatter- 
ing of neutrons by, 47 
capture of neutrons by, 60, 79 
in nucleus, 5, 8 
scattering of, by protons, 64 
scattering of neutrons by, 37 
Pseu do scalar, 107 
Purcell, 15 



1 



Quadrupole moment, 27 
Quadrupole radiation, 13 
Quantitative theory, of exchange 
forces, 87 
of nuclear forces, 23 
Quantum mechanics, 23 



14S 



INDEX 



Rabi, 15, 23, 27 
Radiation, dipote, 13 

interaction of deuteron with, 56 
ootopole, 13 
quadrupole, 13 
Radioactivity, a, lifetimes for, 6 

if, 10, 97 
Ragan, 71 
Rainwater, 1 IS 
Ramsey, 23 
Range, of nuclear forces, 32, 49, 70 

ia relation to depth, 32 
Rari.ta, 27, 59, SO, 73, 76, 77, 87, 88, 

89 
Reactions, nuclear, involving charged 

particles, 8 
Recoil energy, nuclear, 21 
Rectangular potential well, 31, 70 
Reduced mass, 37 
Regularities in stable nuelei, 2 
Relation, between range and depth of 
potential, 32 
between spin and exchange, 84 
Relativistic in variance, 107 
Residual nuclei, 111 
Resonance phenomena, 118 
Resonances, proton, 119 
Richards, 44 
R:idiardson, 101 
Rich-man, 44 
Roberts, 26, 28 
Role of ^-particle, 84 
Rotation of coordinate system, 73 
Rules, selection, Fermi, 99 

Gamow-Teller, 106 
Rutherford, 65 

Sachs, 33 

Saturation, of nuclear forces, 80 

proof of, 84 
Seaixrrg, 111, 123 
Segre, 123 

Scalar in /5-theory, 107 
Scattering, Coulomb, fl'l 

interference between nuclear 
and, 01 

inelastic, energy distribution in, 1 1 4 

nuclear', 64, 114 



Scattering, of neutrons, by heavy nu- 
clei, 7, 1 14 
by ortho-hydrogen, 49 
by para-hydrogen, 49 
by protons, 37, 78 

angular distribution of, at 

high energy, 63 
- experimental results on, 43, 91 
spherical symmetry of, 39 
total cross section for, 40 
by protons bound in molecules, 
47 
of protons by protons, 64 

angular distribution of, 69 
shadow, 114 

spherical symmetry of, 39 
Scherrer, 102, 108 
Schdff, 81 

Schrodinger equation, 31 
Schwinger, 27, 28, 49, 50, 53, 59, 60, 

73, 76, 77, 87, 88, 89 
Selection rules, Fermi, 99 
in /3-dccay, 99 
Gamow-Teller, 106 
Shadow scattering, 114 
Short range of nuclear force, 30, 60 
Skoupp, 20 
Singlet state, of deuteron, 43 

sign of energy in, 45 
Size of nucleus, 6 
Slow neutron cross section, 43 
Spectra, band, 15 
Spectral lines, splitting of, 15 
Spin, 15 

and exchange, relation 

between, 84 
and isotopic spin, 84 
and statistics, 15 
iso topic, 86 
nuclear, 15 
Spin dependence of nuclear force, 45 
Splitting of spectral hues, 15 
Spontaneousemission,offf-particles,6 
of electrons, 10 
of positrons, 10 
Spherical symmetry of scattering, 39 
Stability, of isobars, 11 
of nuclei, 5 



INDEX 



147 



Stable nuclei, regularities in, 2 
States, excited, of nucleus, 111 

of deuteron, 74 
Statistical considerations, 60 
Statistical weights, 01 
Statistics, 10 

Bose, 16 

Fermi, 16 

of neutrons, 1 S 

of protons, 18 

spin and, 15 
Staub, 40 

Stern-Gcrlach experiment, 15 
Stetier, 29 
Stevens, 20, 30 
Sticking probability, 111. 
Structure of spectra, hypcrfine, 15 
Survey of low-energy phenomena of 

deuteron, 62 
Sutton, 53 

Symmetric meson theory, 88 
Symmetry of wave funciton, 68 

Tashelc, 71 
Teller, 49, 50, 106 
Temperature, nuclear, 1 15 
Tensor in /3-theory, 107 
Tensor forces, 28, 75 
Thaxton, 69, 70 
Theory, of atoms, 23 

of exchange forces, 87 
Thermal neutrons, 43 
Thomas, 31 
Tillman, 1 19 
Total cross section for scattering of 

neutrons by protons, 40 
Townsend, 101 

Transitions, allowed, in /3-d i si nteg ra- 
ti on, 99 
lifetimes in, 102 

forbidden, lifetimes in, 105 

7-ra.y, 13 



Transmission coefficient of barrier, 7 
Time, 66 
Tyler, 100 

UMeribeck, 97 

Uncertainty in position, 111 

l/v law, 121 

Van Attn, 29 

Van de Graaff machine, 29 

Variation principle, 80 

Virtual state of deuteron, 15 

VolkajJ, 82 

Wave function, of deuteron, 33 
parity of, 75 
symmetry of, 68 
Wave functions, relativistie eom- 

bi nations of, 107 
Wave length of neutron, 7 
Wave number, de Broglie, 37 
Wei glits, statistical, 61 
Wei.sskopf, 113, 117 
Wells, 20 
Wentz el -B rilloui n-Kramers method, 

7 
White, 66 
Wick, 93 
Width of level, 113 

partial, 1 14 
Wiendenbeck, 29 
Wianer, 28, 30, 43, 44, 45, 52, 73, S4, 

87, 120 
Wigucr argument about short-range 

forces, 30, 66 
Wigner forces, 81 
Williams, 44 
Wilson, 70, 92 

Yukawa, 94 

Zaeharias, 23 



*